Give an example of a beta-lactam antibiotic. How do beta-lactam antibiotics kill bacteria?
Penicillian, amoxicillian and monobactin are all examples of beta-lactamn antibiotics. They attack the structure of the peptydoglycan cell wall to kill bacteria.
What is an "autolysin?" Why do bacteria make autolysins?
Autolysins are bacterial cytoplasmic membrane proteins that break down peptidoglycan. Autolysins give the bacterium some breathing room by breaching the cell wall, which is necessary for the cell wall to lengthen itself in order to divide.
Why aren't gram-negative bacteria killed by vancomysin?
Gram-negative bacteria aren't killed by vancomysin because vancomysin is too big to fit through the outer membrane to the plasmic space.
How have bacteria become resistant to beta-lactam antibiotics?
Bacteria have become resistant because they have mutation capabilities against the antibiotics that destroys the B-lactam. Making an enzyme that destroys penicillian is another way to defeat the antibiotics.
Are any gram-positive bacteria resistant to vancomysin? What might account for this resistance?
Some gram-positive bacteria may be naturally resistant to vancomycin because they hae peptides in their peptidglygan that do not have a terminal D-alanine-D-alanine peptide. Examples are bacteria in the lactobacillus species, such as unpastuerized yogurt.
Name 2 drugs that block bacterial ribosomes. Why does stopping translation kill the cell, even it it's not dividing?
Streptomysin, Zythromax; Proteins do the work of the cell so if you can't produce proteins, you are a gonner!
Why doesn't Zithromax hurt human cells?
Zithromax can't kill human cells because our proteins are just a little different in the ribosomes so it doesn't affect our cells!
Why isn't tetracycline prescribed for pregnant women or young children in first-world countries?
Tetracycline affects the calcium deposits in the bones and teeth to discolor them to a nasty yellow color.
List 2 ways that bacteria can become resistant to antibiotics that stop protein synthesis.
Mutation and proteinsynthesis are 2 ways bacteria become resistant.
How does rifampin work? What infection is it used to treat? Would rifampin kill archaea?
It inhibits the RNA prolymerase, or prevents mRNA from being transcribed. It is used to treat TB, bacterial meningitis. Rifampin would NOT be able kill archaea because archaea are closer to eukaryotic cells than prokaryotic bacteria cells.
Why does isoniazid kill tuberculosis bacteria but not gram-positive or gram-negative bacteria?
Isoniazid doesn't kill gram-positive or gram-negative bacteria because it is an acid-fast bacteria that has mycolic acid that in turn affects the integrity of the cell envelope to make the cell blow up.
What is the target of ciprofloxacin? Why does ciprofloxacin kill bacteria, even when they are not dividing?
What is the difference between an antiseptic and an anitbiotic?
Antibiotics are internally taken whereas antiseptics are more of a general weakener. An antiseptic targets EVERYTHING (fungus, virsus & protazoa) and destroys targets that ALL cells have.
What is the difference between an antiseptic and a disinfectant?
Antiseptics and disinfectants do the exact same thing, the difference is a matter of the concentration. Disinfectants have a high concentration, and are therefore stronger than antiseptics.
What cellular feature do halides destroys? Give an example of halide that is used as an antiseptic, and an example of a halide that is used as a disinfectant?
What cellular feature do alcohols destroy?
Alcohols destroy phospholipids in the cell membrane by dissolving them.
Give an example of an alkylating agent. Why do alkylating agents kill cells?
Formaldehyde. Alkylating agents kill cells by inactivating proteins, which a cell cannot function without.
Why do heavy metals kill cells? Give an example of a heavy metal used as an antiseptic.
Heavy metals kill cells because they denature (change the shape and thus the function) proteins. An example would be silver in catheter plastic.
Can bacteria ever become resistant to disinfectants?
Why is resistance to antibiotics "easier" to evolve than resistance to disinfectants is?


text:
Does anyone else have the occasional epiphany at the post office?
Okay, maybe not. But there's something about me and the P.O.: every time I go there, I get stuck in the longest line possible. No matter if seconds earlier, there's been no one in the entire place. The second I pull in the parking lot, presto-chango, instant crowds.
But back to my epiphany. This morning I was in this hugely long line that was moving way slowly, for two reasons: one of the two clerks working (other windows closed, naturally) was moving at sloth-like speed: I was getting sleepy just watching him. The other factor was a woman at the second window who was getting package weighed under every possible rate to see which was cheapest. I'm surprised her back didn't just burst into flames from all the annoyed looks shot her way.
Anyway, there I am in line, and because this is a friendly sort of town in which I live, the girl in front of me sighs loudly, then rolls her eyes, and says aloud, "God, I gave myself an hour to run all my errands, but now it looks like I'll just be here the whole time." To which I replied, "Oh, it's me. I'm a total jinx. If I'm here, the line is long." She smiled and I went back to staring at the phone cards, thinking this was the end of our exchange. But no. Instead, she then proceeded to tell me pretty much her entire life story. Like how she was finishing up her Master's, and about to move to Greenville to work at Pitt Memorial, because she has this internship, and then she's going to Washington, but today she still has to go get her immunization records from the university AND pick up her boss for a meeting. I mean, this girl was TALKING.
Now. It is a true fact that I share too much. Anyone who has ever met me can tell you that. But today I realized that while I have always felt bad about this personal attribute, it could be much, much worse. All I could think was that to this girl, I was the quiet type. (Little does she know.) And it was kind of nice, actually, that at any given moment on any given day, you can be any given type. If only for as long as it takes to move through a line.
That's all for Deep Thoughts, I promise. At least for today….

redpajama_set_name: RedPajamaCommonCrawl


Back to work, school and the routine, which hopefully includes reading. The lazy days of summer are gone, but there's a virtual American Idol season's-worth of talent in the books out this September.
Many of you expressed the desire to click and buy right away, so you'll now be able to do so when you click on this link. Each month I'll feature an independent bookstore where you can purchase the books. Please send me your suggestions! And you can find the Indie Next booklist at this link Indiebound.org each month, which is a well-curated compendium of books reviewed by book-loving staffers.
This month, I've picked Northshire Bookstore on the main drag in Saratoga, New York (sister store to the amazing one in Manchester, Vermont.) I held a book reading there a few summers ago, and being from upstate New York, the area is near and dear to my heart. What makes Northshire my first indie bookstore spotlight is that they are relatively new to the game. When I heard an indie bookstore was coming back to Saratoga in the age of big box stores and Amazon, it was like watching the movie "Rudy." There was a lot of atta boy-ing. And they've continued to thrive. So if you find yourself in the quaint, racing town of Saratoga, go say hi at Northshire.
In keeping with the spirit of the Bob Woodruff Foundation. I'm adding a category for books that touch on military/veteran or caregiver stories. So many well-written and interesting books are published in this genre and they offer a window onto a world that many of us know little about with today's all volunteer army.
Speaking about our troops – you can earn the Bob Woodruff Foundation $1.00 from now until September 30th simply by posting #give2veterans. You don't even have to open your wallet to help our injured veterans and families, so please pass it on!
So here goes — my humble thoughts and opinions!
I'd read the reviews and was intrigued by the overarching themes of how a woman's career dreams are muted by marriage and motherhood. Then I opened the book and started reading. WHAM– I was totally unprepared for the sweeping scope, breadth, intimacy and depth of this book, reminiscent of a Jonathan Franzen novel. There are so many themes here, from love, betrayal, the parent-child bond, jealousy, favorite children, marital fatigue, what it means to nurse your creative flame while facing extreme head winds. The book is also a story within a story, as some of Joan's writing is featured. The pace of the tale takes a dramatic downshift about two-thirds in, but every time I set it down, I was eager to get back to Joan Ashby.
What happens when a war-tested marine and embedded photojournalist write a book together? It's a well written, compelling back and forth about what it means to go to war and come home. The story of how the men were injured by a bomb blast in Afghanistan and their unlikely friendship will keep you turning pages (and me underlining passages). As the wife of a war correspondent who was injured, there was so much I could relate to, but this doesn't exclude the uninitiated reader.
"Shooting Ghosts" should be mandatory reading for all of us who are "grateful for their service," yet can't quite grasp the tiny, nuanced damage war does to families, loved ones and relationships. The domestic scenes will twist your gut. This book perfectly captures the lonely, aching dissonance many veterans feel when they return to a home front they have dreamed of, that cannot see or understand their wounds. Some of the best, articulate descriptions of post traumatic stress I've ever read.
Ok, the parenting train has pretty much left the station at my house. I've already inflicted terrible damage and bad habits on my children (see reference to vocal fry above) but it may not be too late for your own kids or grandkids. I was intrigued about the promise of the book's ability to help us raise "self-assured children and train their brains for a life of happiness, meaning and success." What's not to love about that?
Being told what NOT to do is also a little more naughty-fun than someone telling you HOW to do it. Social worker Amy Morin lays out her treatise in an easy, sound-bitey way that results in a neat little road map for beginners that should be handed out in maternity wards. It's a form of parenting for dummies that would make a great baby shower gift for tackling the hardest job that comes with absolutely no certification. The father in "My Absolute Darling" (see above) could have used a copy.
If I didn't love this guy so much for being an all-around genuinely good man and generous person, I'd hate this guy for how easily he cranks out thrillers that grab you by the jowls from the first page. He is the only author that got my son reading from dawn till night and his stories have been made into movies and series. He's like a crossover thriller writer. If Jack Reacher is total testosterone and high octane, Coben's characters are more complicated, more the types of people that lurk in our towns. In his latest book, an old high school murder is re-examined to solve a mystery. In the process, the family secrets and lies build on one another like a hurricane out at sea, until they have the power to destroy the entire town. No one does dialogue like my man Harlan.
With the 150th anniversary of "Little Women" around the corner, it's the perfect time to discover the untold story of May Alcott, Louisa's youngest sister and in real life, a visual artist in her own right. From Concord (where my mother lives and the real Alcott house is preserved) to Boston and Europe, she was just hitting her stride as her sister achieved literary fame, portraying May (Amy March) as the "spoiled one" in the classic novel. In this captivating work of historical fiction, Elise Hooper gives readers an imaginative glimpse into the youngest Alcott's artistic pursuits and her side of the sibling rivalry.
When a friend highly recommended this YA book for me I was skeptical, since (full disclosure) I'm not usually a fan of young adult books ( I know, I know, but I did read Twilight!). In fact, I was so confused that I mistakenly bought Emma Donoghue's novel "The Wonder," out in paperback this month and worth the read. FYI- – It's slow burning, beautiful tale of religiosity and family in post-famine Ireland. Was the young Irish lass faking it or not?
But back to this "Wonder," a NYT best seller that will soon be released as a movie. August is born with a tragically disfigured face that he has learned to hide from the world until he enters mainstream school and must re-learn how to navigate in ways that will break your heart open and crochet it back again. Written realistically in August's wise but innocent voice, it reminded me of the book "Room" in the way that you are inside someone's head.
Described to me as "the book that everyone beyond 5th grade should read," it's a tale for the times, to remind all of us to reach for a little more empathy, understanding and kindness. Lecture over.
Based in the same vein as his wonderful debut book "The End of Your Life Book Club," Schwalbe brings a wonderful perspective to curating a list of books that are, in essence, a lesson for living. As a book editor and avid book lover, each chapter is carefully crafted around a specific book. For anyone looking for a "what to read" list, you will find it all here. From classics to children's lit, thrillers and even a cookbook, Schwalbe has woven a play list to help make sense of the world and how we pass through it. I knew he was the man for me when he included my all-time favorite novel "Crossing to Safety" in his first book.


text:
State Secretary reveals plans for October 23rd celebrations
Zoltán Kovács, State Secretary for International Communication and Relations, has highlighted the importance of the upcoming October 23rd celebrations across Hungary.
October 23 1956 Revolution
Zoltán Kovács, State Secretary for International Communication and Relations, has confirmed logistics and highlighted the importance of the upcoming October 23rd celebrations across Hungary.
The operative body responsible for the secure organization of the October 23rd national holiday has authorized the related security plan and celebrations across the country are ready to go.
"October 23rd is the most important manifestation of the Hungarians' desire for freedom and the Hungarian people's battle for independence," Zoltán Kovács said. "On October 23rd, 1956, events occurred with which everyday people, the 'Lads of Pest', set an example not only to Hungarians, but to the whole of the world. Protecting national sovereignty and freedom, and standing up for these, are everyday tasks, and the responsibility of not only politicians, but also everyday people," he added.
The State Secretary highlighted that October 23rd stands as an example of the fact that everyone can take action to achieve freedom even in the most hopeless situations.
Kovács revealed that this year's events are being held under the slogan "The Day of Freedom". On Tuesday, October 22nd, a wreath laying ceremony will be held at the University of Technology's 1956 memorial.
A speech will be given by former Prime Minister Péter Boross. At the event, which will begin at the Budapest University of Technology and Economics at 3 p.m., a speech will also be given by Minister for Innovation and Technology László Palkovics.
Following this, the traditional, period torchlight commemorative procession will take place to the Bem statue, where the Ministry of Human Capacities' Parliamentary State Secretary Bence Rétvári will give a speech.
In the evening, János Csík and Esszencia will give a concert, during and after which a light show will be projected onto the adjacent buildings.
On October 23rd, events will begin at 9 a.m. with the ceremonious hoisting of the flag in front of Parliament on Kossuth Square.
The Holy Crown will be open to the public all day in the Parliament Building, in addition to which family programs will await all visitors on Kossuth Square.
The now traditional Freedom Concert will be held at the Millenáris Park at 7 p.m. in the evening, following which the series of celebratory events will be drawn to a close by a gala concert at the Academy of Music.
Parliament declared October 23rd an official national holiday in 1991, which was later also reinforced by the 2012 Constitution.
How many EU countries have laws on the books that empower the government to mobilise the military in the event of a terror attack?


The 3rd Marine Aircraft Wing is an aviation unit of the United States Marine Corps based at Marine Corps Air Station Miramar, California.
The mission of the 3rd Marine Aircraft Wing is to provide combat ready expeditionary aviation forces capable of short notice world wide deployment to Marine Air-Ground Task Force (MAGTF), fleet and unified commanders.
Our 3rd Marine Air Wing patch is a regulation, full color, embroidered military patch. This is the same patch that is supplied to the Marine Corps and sold in the Marine Corps Base Exchanges. This patch would be a great gift idea and a wonderful addition to your patch collection.


Gloucester Rugby take on Zebre Rugby this Saturday, and full back Rob Cook is looking forward to getting the new Challenge Cup season off and running, accepting the challenge is a tough one but the squad are ready for their title defence.
Saints skipper Dylan Hartley says the side are incredibly frustrated with how their Heineken Cup campaign has unfolded this season, but keen to refocus starting with the return tie with Castres next Sunday.
Lee Dickson sums up the Saints' bitter disappointment having agonisingly missed out on victory against Munster at Thomond Park.The Saints are back at the Gardens next Friday for the visit of the Scarlets in Heineken Cup Round 2.


Members generally feel more ownership and involvement than donors. Members may have an expectation to be asked to contribute other things in addition to money. Some of the things they may contribute are volunteer time, political support, expertise and influence. If these things would benefit your organization, then consider starting a membership program. However, if you really just want money to do your work, then a membership program may add more administration for little added benefit. In that case, look for other ways of showing your gratitude for the money that donors contribute. Donors feel involved and hopefully feel appreciated, but they usually have no expectation of being asked to give anything besides money and feel little ownership of your organization.


text:
Haseldorfer march er den mindste del af de holstenske Elbmarskområder i det nordlige Tyskland. Det ligger nordøst for Elben mellem byen Wedel, Pinnau og gestranden mellem Uetersen og Holm. Den største by i området er Haseldorf; andre bebyggelser er Haselau og Hetlingen. På den modsatte (sydlige) side af Elben ligger byen Stade omkring 20 kilometer vest for Hamborg. Haseldorfer marsk er en del af distriktet Kreis Pinneberg i Slesvig-Holsten. I området lå også landsbyen Bishorst, der forsvandt i Elben. Ved overgangen til Seestermüher Marsch er den næstældste fungerende drejebro i Tyskland bygget i 1887. Indbyggerne i marts måtte håndtere storm oversvømmelser og oversvømmelser.

Begivenheder
I natten den 28. december 1248 ødelagde den såkaldte Allerkindsinsflutte dele af det første digesystem. Store dele af området blev ødelagt.

Ved Hamburger Sturmflut den 16. og 17. februar 1962 blev marsken delvis oversvømmet. Områderne Hetlinger Schanze og Haseldorfer Vorland stod i op til 5,83 meter under vand. Der var dog ingen brudskader i Elb-digerne. Imidlertid viste en række dårligt udførte digeforhøjninger fra slutningen af 1950'erne sig mærkbart; Således faldt den nye digekrone ved Hetlingen simpelthen sammen. På grund af oversvømmelsen ved Pinnau-floden skete der yderligere oversvømmelse i det nærliggende Uetersen. I modsætning til i Hamburg var der ingen dødsfald, men der skete skader på bygninger og infrastruktur på samme måde som i Hamburg-Wilhelmsburg og Georgswerder.

Den 3. januar 1976 blev området igen hjemsøgt af en stormflod i løbet af Capella-orkanerne. Diget ved Hetlingen brød sammen ni steder, og hele marsken blev oversvømmet, hvilket resulterede i skader for millioner. Ved et stort held undgik man dødsfald. Hele 800 familier blev afskåret fra omverdenen. De fleste tab skete blandt husdyr og vilde dyr: 155 stykker kvæg, 610 grise, 280 får og 700 kyllinger blev dræbt. Af vilddyr døde ca. 700 hjorte, 3.000 fasaner og ca. 1.000 harer.

I juli 2002 oversvømmede syndflodsagtige regnvejr Haseldorfer marsk. Der faldt op til 125 liter vand pr. kvadratmeter om 24 timer. Ydermerere strømmede vand til fra gestranden ved Hestlingen. Gartnerier, huse og gader stod under vand. Over 300 brandmænd samt THW og DRK var i brug i dage.

Eksterne henvisninger

 Die Haseldorfer Marsch
 NABU-Naturzentrum Haseldorfer Marsch

Landskaber i Slesvig-Holsten

redpajama_set_name: RedPajamaWikipedia


Geoscience is the science of exploration, discovery, and Earth stewardship. The geosciences address all issues relating to Earth Systems, including the solid Earth, oceans, and atmosphere. The major applications of the geosciences are: exploration and responsible development of natural resources (oil, gas, coal, minerals, construction aggregate, water, soil), preservation of the natural environment, restoration from environmental damage, mitigation of geohazards such as earthquakes and landslides, and exploratory research like the Mars space mission and understanding El Niño.
The B.A. in geosciences at UNCW focuses on the scientific study of Earth; forces acting upon and within it; and the makeup and behavior of the solids, liquids, and gases comprising the planet. This program is designed for students who desire a broad liberal arts education that prepares them for careers such as environmental consulting, energy and resource management, and geosciences education. Because geology continues to become more integrative, technical and global in its approach, the B.A. degree program encourages students to develop basic proficiencies in mathematics, chemistry, physics, and biology, and effective oral and written communication skills including computer literacy.
Scientific communication Knowledge of interconnectedness of different "spheres"


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text:
Watson goes one better, as industry leaders grab limelight
Veteran Wettenhalls driver adds another trophy to mantelpiece, as awards night recognises industry leaders Tony McGrath and Roz Shaw
A veteran truck driver has been recognised for more than five decades of accident-free driving during an awards ceremony that highlighted the contribution of some of the industry's leading representatives.
Wettenhalls truckie William Watson walked away with the National Professional Driver of the Year Award, handed out during the 2012 National Trucking Industry Awards held as part of the Australian Trucking Convention late last week.
The Australian Trucking Association (ATA) cited Watson as "a true professional and elder statesman of the industry" and someone who is always willing to teach younger drivers about the industry.
"He has worked for the Wettenhalls Group for the last 37 years and has recorded more than 50 years of accident-free driving," the ATA says.
Watson won the NatRoad Professional Driver of the Year Award in 2011 and this year was recognised alongside other luminaries such as Hawkins Transport CEO Roz Shaw (pictured) and McGrath Newcastle founder Tony McGrath.
Shaw, whose business encompasses fuel haulage, a Moreton Island tourism venture, truckstops and fuel distribution, won the Trucking Industry Woman of the Year Award.
"To get this award is a very humbling experience. The transport industry is an amazing industry that's undersold to the rest of society," she says.
McGrath, who started his business in 1983, took home the Outstanding Contribution to the Australian Trucking Industry Award for his work in organising the Newcastle and Hunter Region Road Transport Awareness Day and for raising more than $1 million for the Westpac rescue helicopter service.
McGrath is also a director of industry accreditation scheme TruckSafe and is a member of the NSW branch of the ATA. He labelled the award as a proud moment.
"The industry has been my whole life since I left school. It's all I've ever done and all I've wanted to do," he says.
A Linfox linehaul driver, Tim Chellas, was recognised with the Highway Guardian Award for an act of bravery on the Newell Highway last year. Chellas's truck was stopped at a temporary set of lights when he saw a B-double behind him smash into the back of a queue of cars.
The truck went off the road and started to burn, with the driver trapped inside.
"Tim immediately called emergency services and used his onboard fire extinguisher to put out the flames, which unquestionably saved the B-double driver's life," the ATA says.
The group says Chellas was only able to do this because one month earlier he discovered the extinguisher on his truck was defective during a routine check of his vehicle.
"When he opened the bag, he found the extinguisher's trigger mechanism had detached from the tank making it useless. He replaced the fire extinguisher and was able to save the driver's life as a result," the ATA says.
Meanwhile, Star Track won the Training Excellence Award for its national induction training strategy. The program provides all employees with the same induction, which runs over 10 days and involves theory and practical sessions.
"All Star Track drivers are enrolled into a Certificate III of Driving Operations on the commencement of their employment and can then progress into further training over time," the ATA says.
The Port Kembla outfit Cleary Bros was recognised with the John Kelly Memorial Award for excellence in the TruckSafe program. The company, which operates a fleet of more than 150 heavy duty trucks servicing the construction sector, has been a part of the program since 1998.
The ATA says the Don Watson Memorial Award, which highlight's conspicuous achievement in furthering the industry's objectives, was not handed out because the recipient could not attend the awards ceremony.
The ATA says the winner will be announced later this year when the presentation can occur.


text:
Q: Semi-automatic classification plugin - Conversion Problem I'm trying to Preprocess Sentinel-2 Images to use for the classification in the SCP Plugin but I'm hitting a wall at the final step before inputting the Data as Training Input for the Plugin.
I did everything step after step as in this video:
https://www.youtube.com/watch?v=Ceyhm3DlZNY&t=726s
Though when I come to the step (10:17 Preprocess the Data) I get them with RT_clip... in my QGIS but they are simply white or if I copy the style from the clip raster data they turn black.
Here is how it looks when I have one of the clipped visible.

And as an example here with the RT activated with OSM beneath it for reference.

When I look at the TIFF files in my folder they are not only white at all, so I suspect it is some wrong settings in my QGIS. Does somebody have an idea how this is fixable?

redpajama_set_name: RedPajamaStackExchange


Footprints on the Moon is a "filk" band formed by Howard and Raj, playing sci-fi inspired folk music. The band now includes Bert Kibbler, who wrote a geology-inspired song about the boulder that chased Indian Jones.
The band's song was "Hammer and Whip: The Untold Story of Thor vs. Indiana Jones", about the Norse God and whip-carrying archaeologist. Their other songs include "Let's Get Astrophysical" and "Sherlock around the Clock".
The band has performed at Stuart's comic book store, as the house band on "Fun with Flags", and has been hired to play bar-mitzvahs.
The band was formed in "The 2003 Approximation" when Stuart asked Howard and Raj if they knew any musicians who might perform at the show, ideally for free. Raj reminded Howard that they had always talked about performing together and they could play "filk" music, folk music with a sci-fi/fantasy theme.
When Raj mentioned that they would need a cool band name, Howard said he'd been sitting on "Footprints on the Moon" for years. As Howard and Raj got together to write their first song, Raj brought up his idea of a "power ballad, superhero crossover tale of epic proportions" about Thor versus Indiana Jones. Howard began riffing on the theme and soon Raj contributed the hook, "Thor and Dr. Jones. Thor and Dr. Jones. One plays with lightning, the other plays with bones".
After Raj created a Facebook page for the band in "The Earworm Reverberation", Howard was initially annoyed that Raj had acted without consulting him. His concerns were erased when Raj revealed they already had a fan who had commented on their performance at the comic book store.
Howard and Raj decided to write back to their supporter and they quickly became enchanted with their cool follower. When they decided to track him down in real life, they went to a coffee shop they knew he visited. Their excitement faded when they observed their fan doing something gross. As they tried to leave the shop without him realizing, he spotted them and asked if they were "Footprints on the Moon". Howard and Raj said no and kept walking.
When Footprints on the Moon were the house band for Fun with Flags in "The Veracity Elasticity", they performed a new theme song for the podcast. Raj was disgruntled that they didn't get paid for the gig; they weren't even given any food.
Footprints on the Moon were again the house band for a retrospective episode of Fun with Flags in "The Separation Agitation".
Footprints on the Moon were booked to perform at a Bar-Mitzvah in "The Solo Oscillation", but Howard was struggling to keep up with his work, his responsibilities as a father, and preparations for the gig. After Howard suggested that Raj replace him in the band, Raj quickly reached out to Bert Kibbler, who plays the guitar, to join the band. Bert came with his own Indiana Jones-inspired song, told from the point of view of the boulder that chases Indy.
When Bernadette got fed up that Howard was still not very useful around the house, even without the distraction of a band, she told him he could rejoin Footprints on the Moon. The trio then performed at Toby's Bar-Mitzvah, singing Bert's song about the boulder.
Raj and Howard are the house band for an on-location episode of Fun with Flags.
When Howard has to drop out of the band ahead of a gig, Raj recruits geologist Bert to join "Footprints on the Moon". Bert brings his own geology-inspired song about Indiana Jones, told from the perspective of the boulder that chases him. After Howard returns to the band, the newly-formed trio play at a Bar-Mitzvah and perform Bert's song.


text:
Update: PG&E crews start cutting trees near…
Update: PG&E crews start cutting trees near Oroville Cemetery despite protests
Dave Garcia of Oroville sits up in a tree as other protestors gather Monday around trees along Feather River Boulevard near the Oroville Cemetery to try and stop PG&E from cutting them down. - Bill Husa — Mercury-Register
Tree removal specialists working for PG&E remove trees Monday along Feather River Boulevard near the Oroville Cemetery. - Bill Husa — Mercury-Register
Signs hang Monday near the trees along Feather River Boulevard near the Oroville Cemetery PG&E has been seeking to remove. - Bill Husa — Mercury-Register
A bagpiper performs Monday as part of the Save Oroville Trees rally against the planned removal of 13 old sycamore and elm trees in front of Oroville Cemetery on Feather River Boulevard. - Ryan Olson — Mercury-Register
Dave Garcia of Oroville sits up in a tree Monday as he and other protestors gather along Feather River Boulevard near the Oroville Cemetery to try and stop PG&E from cutting 13 ancient trees down. - Bill Husa — Mercury-Register
At lower left Pastor Kevin Thompson of Oroville watches Monday as trees are removed along Feather River Boulevard near the Oroville Cemetery. - Bill Husa — Mercury-Register
Protestors set up a tent Monday after PG&E cut down four of 13 trees along Feather River Boulevard near the Oroville Cemetery. - Bill Husa — Mercury-Register
Kirk Draper of Oroville walks with a "Trees not Stumps" sign Monday as he and other protestors gather around trees along Feather River Boulevard near the Oroville Cemetery. - Bill Husa — Mercury-Register
Tree removal specialists working for PG&E remove trees Monday along Feather River Boulevard near the Oroville Cemetery - Bill Husa — Mercury-Register
Sign marks a tree previously cut by PG&E. - Bill Husa — Mercury-Register
By Barbara Arrigoni, David Pollak and Ryan Olson |
PUBLISHED: January 26, 2015 at 8:06 a.m. | UPDATED: April 21, 2018 at 7:15 a.m.
Oroville >> Despite the efforts of protesters to save 13 trees from being cut down in front of Oroville Cemetery, the chain saws roared to life around 12:10 p.m. Monday and the trees starting coming down.
The removal of four of the trees is another step in ending a stalemate lasting several weeks that pitted PG&E against Oroville citizens who didn't want to see the trees removed. PG&E, worried about the root systems of the trees damaging a gas pipeline in the area, says the work is necessary for safety.
About 60 protesters rallied before dawn Monday as PG&E placed a temporary fence on Feather River Boulevard to keep protesters away. They later added another section of fence to sequester four trees from the protesters, and then the cutting began.
Bill Bynum, spokesman for the Save Oroville Trees citizen group, said he saw crews close the fence gates, cutting off access to four of the trees. There had been two protesters in the now-closed area, but one left believing she could return.
The other protester was told the zone was now a work area. Bynum said she initially refused to leave, but was intimidated and pressured by four or five individuals to depart. PG&E spokesman Donald Cutler later disputed that.
"I feel like crying, like my heart's broken," said Helen Dennis, who has camped out in her wheelchair day after day to save the reportedly historic elm and sycamore trees in front of the Oroville Cemetery. "These trees mean a lot to me. They mean a lot to a lot of people."
Protesters were relieved earlier as it initially appeared the cutting had been averted for another day. Bynum was unhappy that PG&E proceeded ahead of a court hearing Wednesday.
He said PG&E waited until the protesters left to go to work before putting up the fence. He said PG&E lied to police and to the women left protesting.
"They will stoop as low as they can stoop to cut down these historic trees," he said.
"We're here today to take care of this critical safety issue," PG&E's Cutler said. "These trees pose a critical risk to the pipeline and we need to eliminate that risk."
He added the company hopes to finish soon, but only when it is safe for crews and the public.
"We don't want anybody to get hurt," Cutler added. "We've been very careful and taking great strides to make sure it's the best environment it can possibly be."
Protester arrested
Before the cutting began, a group of protesters went over the fence to enter the closed zone. Bynum said they met with Oroville police and ultimately agreed to leave the closed area without resistance, although two were reportedly escorted out of the zone.
"When I jumped the fence, it was because I felt angry, frustrated and betrayed," Bynum said. Protester Alan Cartwright later climbed the fence and re-entered the work area. He was met by Oroville police and was briefly arrested without resistance. He said the Oroville pipeline situation is different than the 2010 incident in San Bruno where a PG&E pipeline exploded, killing eight.
Cartwright said there were safety measures in place to shut the pipe down in case of emergency. He also argued removing the trees could make the pipeline less safe, as the root system would deteriorate.
"I'm mad as hell," Cartwright said. "This is ridiculous."
Need disputed
Community members, lead by Save Oroville Trees, are upset with PG&E's plans to remove the 13 old sycamore and elm trees. The company has been removing 242 trees in the city as part of its Pipeline Pathways project to clear potential hazards along high-pressure gas transmission lines. PG&E officials maintain the trees and roots pose a hazard to the pipeline.
The protesters dispute the need to remove the trees and questioned why PG&E couldn't relocate the pipeline. Some chanted, asking where the previously cut trees went and called for a recall of the City Council.
"We have a mayor and City Council that sold us down the river," protester Linda Draper said.
"I'm disappointed at our leaders," Rev. Kevin Thompson said. "Our elected officials who have turned their back on us. "Not one of them came out to the site to talk with their constituents, to see what could be done, hear our concerns and make observations about where we're at. They all but turned their backs.
"In my opinion, they have sown division in this community and split us down the middle."
Draper and others said the trees were planted in 1887, long before PG&E installed the first pipeline in 1969.
"These trees were here first," Draper said. "They have a right to be here."
In addition to the speeches and protest chants, bagpiper Bill Caspers performed as a low mist drifted over cemetery tombstones in the early dawn light.
PG&E spokesman Paul Moreno said the company had intended to start the work Monday morning.
"We're hoping to conduct this critical safety work, but will only do so when it is safe for our crews and the public," Moreno said before work began.
Regarding where the previously cut trees went, Moreno said the trees were the city's. Most of the trees went to Recology Butte Colusa.
Police stretched
The tree removal is being contested in Butte County Superior Court with two cases to be heard Wednesday. PG&E has a temporary restraining order to prevent Save Oroville Trees protesters from coming within 30 feet of tree cutting. Oroville police had not been enforcing the order because they were not a party to the restraining order matter until last week.
Oroville police were present at times Monday. Chief Bill LaGrone said officers were there to keep the peace as the sides handle the issue. While the protesters have been passionate about saving the trees, he said things have been relatively civil.
"Everybody has been peaceful. No one has been violent," LaGrone said.
The situation has put a strain on police resources and he said no additional staff was available. There were four officers on duty Monday, including LaGrone, and three officers had been on the tree cutting site.
The chief said PG&E's restraining order is specific about applying to protesters entering work zones. He said the work zones can't be established around trees where protesters are present.
Nine trees remained outside of the current work zone Monday. Bynum said Save Oroville Trees was resolved to keep protesters in front of the surviving trees. He said they would camp out at the site to prevent the loss of any more trees, as Rev. Thompson, Virgle Gage and others set up a tent on the sidewalk in front of two trees.
"We're going to stay in front of every single tree, so rather than get arrested we elected to save these nine trees," Bynum said.
Thompson said the group is going to turn the site into a tent city "so they can't hoodwink us like they did today."
Tears streamed down her cheeks as Dennis said she's not leaving.
"If I leave this tree, they're going to take it down," she said. "If they arrest me, OK."
Reach reporter Ryan Olson at rolson@chicoer.com, @NorCalJustice on Twitter, or 896-7763. Reach reporter Barbara Arrigoni at barrigoni@orovillemr.com, @OMRBarbara on Twitter, or 519-8543.
Barbara Arrigoni
David Pollak
Ryan Olson
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The Mardi Himal Trek is a recently initiated route that takes you to a previously undiscovered paradise in the Annapurna Region. For those who want to avoid the crowd, this trek is best suited for you. This trek offers astounding glimpses of Annapurna, Dhaulagiri, Machhapuchare and Manaslu ranges. The prominent features of this trail are Mardi Himal Base Camp, Sidhing Village and Lwang.
After a one hour drive from Pokhara, the trek commences at Kande. The trail first makes a gradual ascent followed by a more challenging climb to Australian Camp at 2060 meters, then slightly downhill to Pothana at 1950 meters and then a climb to Deurali at 2100 meters. Taking a right turn in Deurali the trail passes through lush jungle, first climbing and then undulating on top of a ridge. You come to certain section midway when the trail descends sharply and gradually climbing back up to the ridge till you reach Forest Camp at 2600 meters. On leaving Forest Camp, the climb is quite steep but gets more gradual as you make your way to the ridge. The trail follows Low Camp through the jungle, but the trees become less significant and an hour's hike you emerge from the forest and onto the open ridge to witness breathtaking views of Machhapuchare in front of you and Annapurna South to the left. You can continue trekking for 4-5 hours along the ridge to Mardi Himal Base Camp at 4500 meters and make your way back to High Camp in 4 hours. Although this makes up for a fairly long day, the reward is a viewpoint halfway. Then it is back to Low Camp, followed by a hike to Sidhing, Ghale and Lwang Villages. Head back to Pokhara from Lwang.
Day 01 - Arrival at Kathmandu.
After landing in Kathmandu, a representative from Adventure Steps will whisk you away to your hotel. Along with a welcome drink in the evening, you will be given a comprehensive briefing about your trek.
Day 02 – City sightseeing to UNESCO sites.
Today you will be taken on a full day of sightseeing to visit UNESCO World Heritage Sites like Kathmandu Durbar Square, Boudhanath, Swayambunath ( The Monkey Temple ) and Pashupatinath all within the Kathmandu Valley. A well versed tour guide will accompany you for a guided tour. The evening can be spent browsing and wandering around the vibrant streets of Thamel, its excellent restaurants and souvenir shops.
Day 03 - Drive or Fly to Pokhara.
You can either take the 6-7 hour scenic drive to Pokhara or hop onto the 25 minute domestic flight to Pokhara. However, the 200 km drive to Pokhara has its rewards of views of terraced farms, winding rivers and lush forests. On reaching Pokhara and after checking into your hotel, you can take a stroll along the banks of the river or go boating and enjoy the views of the Annapurna ranges. If you are driving to Pokhara, lunch will be en route.
Day 04 - Pokhara to Dhampus (1600 metres ).
Day 05 - Dhampus to Forest Camp ( 2480 meters ).
The trek today is approx. 6 hours. Most of this section is through enchanting forests of birch trees, flaming rhododendrons, oak, maple, hemlock and daphnia; a naturalist's haven. Camping overnight here is all the more appealing after a arduous hike.
Day 06 - Forest Camp to Low Camp ( 3150 meters ).
As we ascend, there is a marked change in the vegetation with mosses, lichen, ferns and exotic species of orchids that cling unto hardwood trees. The gaps through the trees frame peaks like Mt. Machhapuchare, Mardi Himal and the Annapurna range and form a picturesque backdrop in all its splendor.
Day 07- Low Camp to High Camp (4100 meters ).
This day will be set aside for rest as we need to revitalize ourselves for our onward journey. Short hikes here and there can be compensated with more wonders.
Day 09 - High Camp to Sidhing Village ( 1280 meters ).
After breakfast, we make a steady descent down through the cool cover of the forests till we come out unto terraced fields. This is follows a different route to Mardi Khola. As we traverse along the Mardi Khola from the hill settlement of Sidhing, descending through graceful fields, we will pass quaint farms and cross suspension bridges. This region is one of the most scenic and beautiful example of the Nepalese countryside. Overnight stay at camp.
Day 10 - Sidhing Village to Pokhara</strong.
A brief hike and skirting an outcrop takes us to our pick up point where our transport awaits. We will make it back to Pokhara for an appetizing lunch. The soothing climate and the scenic picture perfect character of this place does wonders for your senses. An opportunity to explore the Mahendra Cave, Davis Fall and various placid lakes of Pokhara to add a feather to your cap of picturesque adventure.
Day 11- Drive to Kathmandu OR an Optional 25 minute flight.
Once again a scenic drive back to Kathmandu alongside the winding rivers. Last minute souvenir hunting in Kathmandu and an evening to reflect on your Mardi Himal adventure. Or a 25 minute flight back to Kathmandu.Overnight at hotel.
We will see you off at the Tribhuvan International Airport on your departure to your home country.
Airports pick up & drop off.
Round Trip bus ticket to Kathmandu – Pokhara.
Pokhara to Phedi drop off by car and end of the day pick up from Sidhing Village.
Accommodation in Hotel Himalayan Oasis. B/B Basis for three nights.
Accommodation in Pokhara for two nights.
Accommodation in Tea house during the trek.
Three meals Breakfast, Lunch and dinner included during the trek.
Professional, English-speaking, government-licensed holder guide.
TIMS, Trekking permit and National Park Entry Fee.
First Aid kit and required medicines carried by the guide.
Personal expenses like Beverages, Snacks ,Hot Shower, Wi-fi etc.
Extra night accommodation in case of early arrival or late departure.
If the trip is affected due to circumstances like natural calamities, strike etc. which beyond our control.


text:
\section*{Introduction}

The study of quiver Hecke algebras is a recent development in representation theory, dually motivated from studying representations of cyclotomic Hecke algebras and Khovanov-Lauda-Rouquier (KLR) algebras. Whereas cyclotomic Hecke algebras, being related to symmetric groups and classical Iwahori-Hecke algebras, have been studied for some time, KLR algebras are relative newcomers, and the two ostensibly different families were linked by groundbreaking work of Brundan and Kleshchev \cite{BK:GradedKL}. Because of this deep and meaningful link, it is becoming customary to refer to KLR algebras as quiver Hecke algebras. In recent years much work has been done to study these algebras in their own right. Mathas \cite{MathasSurvey} gives a particularly erudite survey of such developments.

Quiver Hecke algebras are associative graded algebras whose presentation by generators and relations, and whose grading, depends on the data of a quiver. Particularly interesting is the case when the quiver is simply laced, as quotients of these algebras are isomorphic to cyclotomic Hecke algebras, endowing these algebras with a $\mathbb{Z}$-grading. In this paper, we define a new family of algebras called alternating quiver Hecke algebras. For quiver Hecke algebras whose quivers are simply laced, it is possible to define a homogeneous involution on the quiver Hecke algebra; studying the fixed-point subalgebra under this involution gives rise to our family of algebras. We discuss these algebras using a version of Clifford theory for associative algebras, and construct a homogeneous basis and a presentation by homogeneous generators and relations which are reminiscent of Khovanov and Lauda \cite{KhovLaud:diagI} and Rouquier's \cite{Rouq:2KM} theorems for quiver Hecke algebras. Cyclotomic quotients of these algebras are studied in \cite{BM2} and \cite{BM}.

This paper is organised as follows. We start by giving the definition of quiver Hecke algebras for arbitrary quivers. Then we discuss the Clifford theory for associative algebras that will give the technical mechanism for most of our proofs; this relies on the construction of the opposite quiver. In Chapter 3 we discuss alternating quiver Hecke algebras, giving a basis theorem for this new family of algebras. Finally, in Chapter 4 we prove the main result of this paper, giving a KLR-style presentation for alternating quiver Hecke algebras.

{\bf Acknowledgements.} The author was supported by an Australian Postgraduate Award.
The author would like to thank his supervisor Andrew Mathas for his guidance throughout his PhD candidature, as well as his thesis examiners for suggesting some improvements to the arguments in this paper. A slightly modified version of this paper appears as Chapter 5 of the author's PhD thesis \cite{BoysThesis} at the University of Sydney.

\section{Quiver Hecke algebras and opposite quivers}

In this chapter we define quiver Hecke algebras, and give some basic properties. These algebras were introduced by
Khovanov and Lauda~\cite{KhovLaud:diagI} and Rouquier~\cite{Rouq:2KM}.

Let $\Gamma$ be a quiver with vertex set $I$. Following Kac~\cite{Kac}, to the simply-laced quiver $\Gamma$ we attach the usual
Lie theoretic data of the positive roots $\set{\alpha_i\mid i\in I}$, the
fundamental weights $\set{\Lambda_i|i\in I}$, the non-degenerate
pairing $(\Lambda_i,\alpha_j)=\delta_{ij}$, for $i,j\in I$, and the
{Cartan matrix} $C=(c_{ij})_{i,j\in I}$ where
\begin{equation}\label{cartanmatrixdef}
c_{ij}=\begin{cases}
        2,&\text{if }i=j,\\
        -1,&\text{if $i\leftarrow j$ or $i\rightarrow j$},\\
        0,&\text{otherwise.}
      \end{cases}
\end{equation}
Let $P^+=\bigoplus_{i\in I}\mathbb{N}\Lambda_i$ and $Q^+_\Gamma=\bigoplus_{i\in I}\mathbb{N}\alpha_i$.
The {height} of $\beta=\sum_i a_i\alpha_i\in Q^+_\Gamma$ is the
non-negative integer $\height\beta=\sum_ia_i\in\mathbb{N}$. Fix $n\ge0$ and let
$Q^+_n=\set{\beta\in Q^+_\Gamma\mid\height\beta=n}$. For $\beta\in Q^+_\Gamma$ let
\begin{equation}\label{Ialphadef}
I^\beta=\set{\mathbf{i}=(i_1,\dots,i_n)\in I^n|\beta=\alpha_{i_1}+\dots+\alpha_{i_n}}.
\end{equation}
In this paper this data plays only a superficial role in describing the combinatorics of the
grading on the algebras that we consider.

Let $\mathcal{Z}$ be an arbitrary unital associative integral domain.

\begin{Definition}[Khovanov and Lauda \cite{KhovLaud:diagI}
                   and Rouquier~\cite{Rouq:2KM}]
 \label{D:klrdef}
 Let $\Gamma$ be a simply-laced quiver with vertex set $I$ and suppose that  $\beta\in Q^+_\Gamma$.
  The
  {quiver Hecke algebra} $\mathcal{R}_\beta(\Gamma)=\mathcal{R}_\beta^\Lambda(\Gamma,\mathcal{Z})$ is the unital associative $\mathcal{Z}$-algebra  with generators
  \begin{equation*}
    \{\psi_1,\ldots,\psi_{n-1}\}\cup \{y_1,\ldots,y_n\}
         \cup \{e_\Gamma({\mathbf{i}})\mid {\mathbf{i}}\in I^\beta\}
  \end{equation*}
  and relations
  {\setlength{\abovedisplayskip}{2pt}
   \setlength{\belowdisplayskip}{1pt}
  \begin{xalignat*}{3}
    e_\Gamma({\mathbf{i}})e_\Gamma({\mathbf{j}})&= \delta_{{\mathbf{i}}_\Gamma{\mathbf{j}}_\Gamma}e_\Gamma({\mathbf{i}}),
        &\textstyle\sum_{{\mathbf{i}}\in I^\beta}e_\Gamma({\mathbf{i}})&= 1,\\%\label{E:top3}
  y_re_\Gamma({\mathbf{i}})&= e_\Gamma({\mathbf{i}})y_r,& \psi_re_\Gamma({\mathbf{i}})&= e(s_r\cdot{\mathbf{i}}_\Gamma)\psi_r,& y_ry_s&= y_sy_r,

  \end{xalignat*}
  \begin{xalignat*}{2}
    \psi_ry_{r+1}e_\Gamma({\mathbf{i}})&= (y_r\psi_r+\delta_{i_r i_{r+1}})e_\Gamma({\mathbf{i}}),
  & y_{r+1}\psi_re_\Gamma({\mathbf{i}})&= (\psi_ry_r+\delta_{i_r i_{r+1}})e_\Gamma({\mathbf{i}}),
  \end{xalignat*}
  \begin{align*}
  \psi_ry_s&= y_s\psi_r,&\text{if }s\neq r,r+1,\\
  \psi_r\psi_s&= \psi_s\psi_r,&\text{if }|r-s|>1,
  \end{align*}
  \begin{align}
  \psi_r^2e_\Gamma({\mathbf{i}})&= \begin{cases}
      0,&\text{if }i_r=i_{r+1},\\
      (y_r-y_{r+1})e_\Gamma({\mathbf{i}}),&\text{if }i_r\to i_{r+1},\\
      (y_{r+1}-y_r)e_\Gamma({\mathbf{i}}),&\text{if }i_r\leftarrow i_{r+1},\\
      e_\Gamma({\mathbf{i}}),&\text{otherwise},
    \end{cases}\label{E:psirsquared}\\
  \psi_r\psi_{r+1}\psi_re_\Gamma({\mathbf{i}})&=\begin{cases}
    (\psi_{r+1}\psi_r\psi_{r+1}-1)e_\Gamma({\mathbf{i}}),&\text{if }i_r=i_{r+2}\to i_{r+1},\\
    (\psi_{r+1}\psi_r\psi_{r+1}+1)e_\Gamma({\mathbf{i}}),&\text{if }i_r=i_{r+2}\leftarrow i_{r+1},\\
    \psi_{r+1}\psi_r\psi_{r+1}e_\Gamma({\mathbf{i}}),&\text{otherwise,}
  \end{cases}\label{E:deformedbraid}
  \end{align}
  }%
  for ${\mathbf{i}},{\mathbf{j}}\in I^\beta$ and all admissible $r$ and $s$. If $n\ge0$
  then the {quiver Hecke algebra} is the algebra $\mathcal{R}_n=\mathcal{R}_n(\Gamma,\mathcal{Z})=\bigoplus_{\beta\in Q^+_\Gamma}\mathcal{R}_\beta(\Gamma,\mathcal{Z})$.
\end{Definition}

One can check easily that the relations in \autoref{D:klrdef} are homogeneous with respect
to the following $\mathbb{Z}$-valued degree function:
\begin{align}
  \deg e_\Gamma({\mathbf{i}})&= 0, &\text{for all }{\mathbf{i}}\in I^n,\nonumber\\
  \deg y_r&= 2,&\text{for }1\le r\le n,\label{degfunc}\\
  \deg \psi_re_\Gamma({\mathbf{i}})&= -c_{i_r,i_{r+1}},&\text{for $1\le r<n$ and ${\mathbf{i}}\in I^n$}.
  \nonumber
\end{align}
Therefore, $\mathcal{R}_n(\Gamma)$ is a $\mathbb{Z}$-graded algebra.

Using \eqref{Ialphadef}, if we define
\begin{equation}\label{blockidem}
e_\Gamma(\alpha)=\sum_{{\mathbf{i}}\in I^\beta}e_\Gamma({\mathbf{i}})
\end{equation}
for $\beta\in Q^+_\Gamma$
then the algebras $\mathcal{R}_\beta=e_\Gamma(\beta) \mathcal{R}_ne_\Gamma(\beta)$ are actually the {\em blocks} of $\mathcal{R}_n$ \cite[Corollary 2.11]{KhovLaud:diagI}

For each permutation $\omega\in\Sn$, fix a reduced expression $\omega=s_{i_1}s_{i_2}\cdots s_{i_n}$ and define
\[
\psi_\omega=\psi_{i_1}\psi_{i_2}\cdots\psi_{i_n}.
\]
Importantly, because the braid relations for the elements $\{\psi_1,\psi_2,\ldots,\psi_{n-1}\}$ are more complicated than the symmetric group braid relations, the elements $\psi_\omega$ {\em do} depend on the choice of reduced expression. For $\mathbf{a}=(a_1,a_2,\ldots,a_n)\in \mathbb{N}^n$, write $y^{\mathbf{a}}\in \mathcal{R}_n$ for the monomial  $y_1^{a_1}y_2^{a_2}\cdots y_n^{a_n}$. The following theorem appears as \cite[Theorem 2.5]{KhovLaud:diagI} and \cite[Theorem 3.7]{Rouq:2KM}.

\begin{Theorem}[Basis theorem for quiver Hecke algebras]\label{qHbasis}
Let $\beta\in Q^+$. Then $\mathcal{R}_\beta(\Gamma)$ is a free $\mathcal{Z}$-algebra with homogeneous basis
\[
\{\psi_\omega y^{\bf a}e_\Gamma({\mathbf{i}})\mid \omega\in\Sn,\;{\bf a}\in \mathbb{N}^n,\; {\mathbf{i}}\in I^\beta\}.
\]
\end{Theorem}

In this paper we will be interested in the following construction on quivers.

\begin{Definition}
For a quiver $\Gamma$ with vertex set $I$, let $\Gamma'$ be the {\em opposite quiver} whose vertex set is $I$ and which has edges $j\to i$ whenever there is an edge $i\to j$ in $\Gamma$, for $i,j\in I$.
\end{Definition}

\begin{Remark}
Note that a quiver may be isomorphic to its own opposite under a bijection $\tau:I\to I$. In this case there will be edges $\tau(i)\to \tau(j)$ in $\Gamma'$ for every edge $i\to j$ in $\Gamma$. This is why we label idempotents $e_\Gamma({\mathbf{i}})$ with the quiver $\Gamma$
\end{Remark}

\begin{Example}\label{typeAex}
Fix an integer $e\in\{0,3,4,5,\ldots\}$ and take $\Gamma$ to be
the quiver $\Gamma_e$ with vertex set $I=\mathbb{Z}/e\mathbb{Z}$ and edges $i\to i+1$, for $i\in
I$.  That is, $\Gamma_e$ is the infinite quiver of type $A_\infty$ if
$e=0$ and the finite quiver of Dynkin type $A_{e-1}^{(1)}$ if $e\geq 3$.
Notice that $\Gamma_e'$ is isomorphic to $\Gamma_e$ under the bijection
\begin{align*}
\tau_e:I&\to I\\
i&\mapsto -i\quad\mbox{mod $e$}.
\end{align*}
The results in this paper in the particular case when $\Gamma=\Gamma_e$ are discussed at length in \cite[Chapter 5]{BoysThesis}. This is how the result for general simply-laced $\Gamma$ may be motivated. Throughout this paper we give examples of how our general results work for this special case.
\end{Example}

\section{Clifford theory for associative algebras}

In order to deduce results about our fixed-point subalgebras from the corresponding results for the full algebra, we will use the language of Clifford theory. Clifford theory was initially developed to study the representations of normal subgroups of finite groups \cite{Clifford}. Here we adapt it to cover associative algebras with a $C_2$-graded Clifford system; details in the finite group case can be found in \cite{CRR}, and a slightly different and more general treatment for associative algebras is given in \cite{RR}. Below we write the two elements of the cyclic group $C_2$ as $\{+,-\}$, where signs multiply according to the usual rules. Again, let $\mathcal{Z}$ be an arbitrary associative unital integral domain.

\begin{Definition}\label{clifforddef}
Let $A$ be a $\mathcal{Z}$-algebra. A {\em $C_2$-graded Clifford system} for $A$ is a family $\{A_s\mid s\in C_2\}$ of two $\mathcal{Z}$-submodules of $A$ such that
\begin{itemize}
\item[(i)] $A_sA_t=A_{st}$, for $s,t\in C_2$;
\item[(ii)] there exists a distinguished central element $\varepsilon\in A$ such that $\varepsilon^2=1$ and $A_+=\varepsilon A_-$;
\item[(iii)] $A=A_+\oplus A_-$; and
\item[(iv)] $1\in A_+$.
\end{itemize}
$A_+$ is called the {\em even} part of the algebra; $A_-$ is called the {\em odd} part.
\end{Definition}

Clifford theory allows us to give a neat description of the representation theory of the subalgebra $A_+$ given knowledge of the representation theory of $A$. Specifically, for an $A_+$-module $M$, we may twist the $A_+$-action by $\varepsilon$ to define the module $M^\varepsilon$, which is $M$ as an $\O$-module and where $a\in M$ acts as
\[
a\cdot m=\varepsilon a \varepsilon m.
\]
Using Definition \ref{clifforddef}(ii) to realise $C_2$ as $\{1,\varepsilon\}$, the {\em inertia group} of $M$ is
\[
\mathcal{I}(M)=\{x\in C_2\mid M^x\cong M\}\unlhd C_2.
\]
The size of the inertia group (either 1 or 2) determines the behaviour of an irreducible representation $N$ restricted from $A$ to $A_+$, which we denote by $\Res^A_{A_+}\!N$, in the following sense. We refer the reader to \cite[pp344--345]{CRM} for a proof of the following result, noting that inverting 2 is necessary to prove the direct sum decomposition in (ii).

\begin{Proposition}[Clifford's theorem for $C_2$-graded associative algebras]\label{clifford}
Let 2 be invertible in $\mathcal{Z}$ and let $A$ be an associative $\mathcal{Z}$-algebra with a $C_2$-graded Clifford system.  Let $N$ be an irreducible $A$-module. Then
\begin{itemize}
\item[(i)] If $\mathcal{I}(\Res^A_{A_+}\!N)=C_2$ then $\Res^A_{A_+}\!N\cong \Bigl(\Res^A_{A_+}\!N\Bigr)^\varepsilon$ is an irreducible $A_+$-module.
\item[(ii)] If $\mathcal{I}(\Res^A_{A_+}\!N)=1$ then $\Res^A_{A_+}\!N=M_+\oplus M_-$ is the direct sum of two irreducible $A_+$-modules, related under the conjugation map (i.e. $M_+=M_-^\varepsilon$).
\end{itemize}
Moreover, {\em all} irreducible $A_+$-modules arise in one of these two ways.
\end{Proposition}

\section{Alternating quiver Hecke algebras}

Given a result concerning symmetric groups, it is natural to ask what this result implies for its index-2 subgroup, the alternating group. It is this construction and line of enquiry which motivates our definition of alternating quiver Hecke algebras, which are related to quiver Hecke algebras in much the same way. Indeed, as the alternating group algebra is nothing more than the subgroup of the symmetric group algebra of points fixed by the sign map \cite[\S2.1]{James}, our alternating quiver Hecke algebras will be subalgebras of fixed points under an analogous map, this time a homogeneous involution. Recall the definition of the quiver Hecke algebra $\mathcal{R}_n(\Gamma)$ from \S1. We now extend the original definition of Kleshchev, Mathas and Ram \cite{KMR:UniversalSpecht} of the graded sign map.

\begin{Definition}\label{gsgndef}
The {\em graded sign map} $\mathtt{sgn}:\mathcal{R}_n(\Gamma)\to\mathcal{R}_n(\Gamma')$ is the map defined on generators as
\[
e_\Gamma({\mathbf{i}})\mapsto e_{\Gamma'}({\mathbf{i}}),\qquad y_r\mapsto -y_r,\qquad \psi_r\mapsto -\psi_r.
\]
\end{Definition}

\begin{Proposition}\cite{KMR:UniversalSpecht}
The graded sign map is a well-defined homogeneous algebra homomorphism $\mathcal{R}_n(\Gamma) \to \mathcal{R}_n(\Gamma')$.
\end{Proposition}
\begin{proof}
That the map is homogeneous is clear from its definition; checking it is a well-defined algebra homomorphism amounts to checking it respects the list of relations in Definition \ref{D:klrdef}. Once the notation is correctly interpreted, remembering that in the assignment $e_\Gamma({\mathbf{i}})\xrightarrow{\mathtt{sgn}} e_{\Gamma'}({\mathbf{i}})$, the ${\mathbf{i}}$ on the right is a sequences of vertices in the opposite quiver, whose edges are reversed, this is a straightforward exercise which we leave to the reader.
\end{proof}

\begin{Definition}
Let $\Gamma$ be a simply-laced quiver and $\Gamma'$ its opposite. We write $\tau:I\to I$ for the map for which $i\to j$ in $\Gamma$ if and only if $\tau(j)\to \tau(i)$ in $\Gamma'$. $\tau$ is called the {\em reversal map}.
\end{Definition}

We can extend the reversal map $\tau$ to $Q^+_\Gamma$ as follows. Let $\alpha=\alpha_{i_1}+\alpha_{i_2}+\ldots+\alpha_{i_n}\in Q^+_\Gamma$; then define $\tau(\alpha)$ to be the root $\alpha_{\tau(i_1)}+\alpha_{\tau(i_2)}+\ldots+\alpha_{\tau(i_n)}$. The following lemma gives the consequences of our notation.

\begin{Lemma}\label{alphalemma}
There is an isomorphism of algebras $\mathcal{R}_\alpha(\Gamma)\cong \mathcal{R}_{\tau(\alpha)}(\Gamma')$. Under this isomorphism $e_\Gamma({\mathbf{i}})$ is identified with $e_{\Gamma'}(\tau({\mathbf{i}}))$.
\end{Lemma}

Notice that, on the level of blocks, the $\mathtt{sgn}$ map takes $\mathcal{R}_\alpha(\Gamma)$ to $\mathcal{R}_{\alpha}(\Gamma')$ and so is an involution on the direct sum algebra $\mathcal{R}_\alpha(\Gamma)\oplus \mathcal{R}_{\alpha}(\Gamma')$. Where it is convenient, we often use Lemma \ref{alphalemma} to abuse notation without comment in the sequel. We are now ready to define alternating quiver Hecke algebras, which we will study for the remainder of this paper.

\begin{Definition}\label{alphaequiv}
We write $[\alpha]$ for the equivalence class of $\alpha\in Q^+_\Gamma= Q^+_{\Gamma'}$ under the equivalence relation $\sim$ generated by $\alpha\sim \beta$ if $\beta=\tau(\alpha)$. We write $(Q^+_{\Gamma})_\tau$ for the set of all equivalence classes.
\end{Definition}

\begin{Definition}
For a simply-laced quiver $\Gamma$ and $\alpha\in Q_\Gamma^+$, the {\em alternating quiver Hecke algebra} is the fixed-point subalgebra
\[
\mathcal{R}_\alpha(\Gamma/\Gamma')^\mathtt{sgn}=\Bigl[\mathcal{R}_\alpha(\Gamma)\oplus \mathcal{R}_\alpha(\Gamma')\Bigr]^\mathtt{sgn}
\]
of the quiver Hecke algebra under the graded sign map. The alternating quiver Hecke algebra $\mathcal{R}_n^\mathtt{sgn}$ is the algebra
\begin{equation}\label{altqhdecomp}
\mathcal{R}_n^\mathtt{sgn} =\bigoplus_{[\alpha]\in (Q^+_\gamma)_\tau}\mathcal{R}_\alpha(\Gamma/\Gamma')^\mathtt{sgn}
\end{equation}
Note that this decomposition is independent of our choice of equivalence class representative $\alpha\in[\alpha]$ by Lemma \ref{alphalemma}; by the same lemma we can identify $\mathcal{R}_n^\mathtt{sgn}$ with a subalgebra of $\mathcal{R}_n$.
\end{Definition}

\begin{Remark}
It is important to note the summands appearing on the right-hand side of \eqref{altqhdecomp} in general are not the {\em blocks} of the alternating quiver Hecke algebras, as they are not guaranteed to be indecomposable (although most of them are). It would be an interesting problem to classify the blocks of alternating quiver Hecke algebras.
\end{Remark}

We also want to work with equivalence classes of $I^n$ under the involution on residue sequences ${\mathbf{i}}$ induced by the reversal map $\tau$.  More precisely, for ${\mathbf{i}},{\mathbf{j}}\in I^n$ let $\sim$ be the equivalence relation on $I^n$ generated by ${\mathbf{i}}\sim {\mathbf{j}}$ if ${\mathbf{i}}=\tau({\mathbf{j}})$, where $\tau({\mathbf{i}})=(\tau(i_1),\tau(i_2),\ldots,\tau(i_n))$ for ${\mathbf{i}}=(i_1,\ldots,i_n)$. From this we obtain a partition of the set $I^n$ into equivalence classes of size 1 or 2, depending on whether or not ${\mathbf{i}}=\tau({\mathbf{i}})$. We denote the equivalence class containing a sequence ${\mathbf{i}}$ by $[{\mathbf{i}}]$. We denote the set of equivalence classes by $I^n_\tau$ and in each equivalence class we choose a representative ${\mathbf{i}}_\tau\in[{\mathbf{i}}_\tau]$. \label{eiplusdef} Similarly we write $I^\alpha_\tau$ for the set of $\tau$-equivalence classes in $I^\alpha$, noting that $I^\alpha_\tau = I^{\tau(\alpha)}_\tau$.

Using these equivalence classes and distinguished elements, we now define an element $\varepsilon\in \mathcal{R}_n$ which will be very important in studying alternating quiver Hecke algebras and their cyclotomic quotients.
Define $\varepsilon$ to be the element
\begin{equation}\label{varepdef}
\varepsilon=\sum_{\substack{[{\mathbf{i}}]\in I^n_\tau\\ {\mathbf{i}}_\tau\in [{\mathbf{i}}]}}\Bigl(e_\Gamma({\mathbf{i}}_\tau)-e_{\Gamma}(\tau({\mathbf{i}}_\tau))\Bigr)\in \mathcal{R}_n.
\end{equation}
Notice that by Lemma \ref{alphalemma} we can also identify $\varepsilon$ with an element of the direct sum algebra $\mathcal{R}_n(\Gamma)\oplus \mathcal{R}_n(\Gamma')$. It also has the important properties
\[
\mathtt{sgn}(\varepsilon)=-\varepsilon\quad\mbox{and}\quad \varepsilon^2 = 1.
\]

Recall the idempotents $e_\Gamma(\alpha)$ from \eqref{blockidem}, which define the blocks of quiver Hecke algebras. We now define new idempotents
\begin{equation}\label{altblockidem}
e_{[\alpha]}=e_\Gamma(\alpha) + e_{\Gamma'}(\alpha),
\end{equation}
noting that since $[e_\Gamma(\alpha)]^\mathtt{sgn}=e_{\Gamma'}(\alpha)$, $e_{[\alpha]}$ is $\mathtt{sgn}$-invariant; in particular by Lemma \ref{alphalemma}, $e_{[\alpha]}$ is independent of the choice of equivalence class representative of $[\alpha]$.
Finally, for an equivalence class $[{\mathbf{i}}]\in I^n_\tau$ we write
\begin{equation}\label{esquaredef}
e[{\mathbf{i}}]=e_\Gamma({\mathbf{i}})+e_{\Gamma'}({\mathbf{i}})
\end{equation}
which again does not depend on the choice of equivalence class representative.

We now give a basis theorem for alternating quiver Hecke algebras analogous to Theorem \ref{qHbasis} for quiver Hecke algebras. Recall that $(Q^+_{\Gamma})_\tau$ is the set of all equivalence classes $[\alpha]$. For ${\bf a}=(a_1,a_2,\ldots,a_n)\in\mathbb{N}^n$, let $\left|\mathbf{a}\right|=\sum_{i=1}^n a_i$.

\newcommand\SetBox[3][60]{\Bigg\{\ #2\ \Bigg| \ \vcenter{\hsize#1mm\centering #3}\Bigg\}}

\begin{Theorem}[Basis theorem for alternating quiver Hecke algebras]\label{aqHbasis}
Let $\Gamma$ be a simply-laced quiver and let 2 be invertible in $\mathcal{Z}$. For $[\alpha]\in (Q^+_{\Gamma})_\tau$ the alternating quiver Hecke algebra $\mathcal{R}_{\alpha}(\Gamma/\Gamma')^\mathtt{sgn}$ has homogeneous basis
\[
\SetBox[75]{\psi_\omega y^{\mathbf{a}}\varepsilon^b e[{\mathbf{i}}]}{$\omega\in\Sn,\quad\mathbf{a}\in\mathbb{N}^n,\quad [{\mathbf{i}}]\in I^{\alpha}_\tau$, $b\in \{0,1\},\quad \ell(\omega)+\left|\mathbf{a}\right|+b\equiv 0 \; \mbox{mod 2}$}
\]
\end{Theorem}

\begin{proof}
Let $[\alpha]\in (Q^+_{\Gamma})_\tau$. To observe that the specified set spans $\mathcal{R}_{\alpha}(\Gamma/\Gamma')^\mathtt{sgn}$, let us write an arbitrary element of $\mathcal{R}_\alpha(\Gamma)\oplus \mathcal{R}_{\alpha}(\Gamma')\cong \mathcal{R}_\alpha(\Gamma)\oplus\mathcal{R}_{\tau(\alpha)}(\Gamma)$ as a finite sum
\[
x=\sum_{\omega,\mathbf{a},{\mathbf{i}}}\lambda_{\omega,\mathbf{a},{\mathbf{i}}}\psi_\omega y^{\mathbf{a}}e_\Gamma({\mathbf{i}}),
\]
for $\omega\in\Sn$, $\mathbf{a}\in \mathbb{N}^n$, ${\mathbf{i}}\in I^\alpha\cup I^{\tau(\alpha)}$ and $\lambda_{\omega,\mathbf{a},{\mathbf{i}}}\in \mathcal{Z}$ using Theorem \ref{qHbasis} (note we are implicitly using Lemma \ref{alphalemma} here to write this as an element of $\mathcal{R}_n(\Gamma)$). In order that $x$ be an element of $\mathcal{R}_{\alpha}^\mathtt{sgn}$ we require $x^\mathtt{sgn}=x$, i.e. that
\[
\sum_{\omega,\mathbf{a},{\mathbf{i}}}(-1)^{\ell(\omega)+\left|\mathbf{a}\right|}\lambda_{\omega,\mathbf{a},\tau({\mathbf{i}})}\psi_\omega y^{\mathbf{a}}e_\Gamma({\mathbf{i}})=\sum_{\omega,\mathbf{a},{\mathbf{i}}}\lambda_{\omega,\mathbf{a},{\mathbf{i}}}\psi_\omega y^{\mathbf{a}}e_\Gamma({\mathbf{i}})
\]
which implies that the given elements span by applying Theorem \ref{qHbasis} and equating coefficients. For linear independence, take a linear combination
\[
\sum_{[{\mathbf{i}}]\in I^{\alpha}_\tau}\Bigl(\sum_{\substack{\omega\in\Sn,\mathbf{a}\in\mathbb{N}^n\\ \ell(\omega)+\left|\mathbf{a}\right|\equiv 0 \; \text{mod 2}}}\lambda_{\mathbf{a},\omega,{\mathbf{i}}}\psi_\omega y^{\mathbf{a}}e[{\mathbf{i}}]+\sum_{\substack{\omega\in\Sn,\mathbf{a}\in\mathbb{N}^n\\ \ell(\omega)+\left|\mathbf{a}\right|\equiv 1 \; \text{mod 2}}}\lambda_{\mathbf{a},\omega,{\mathbf{i}}}\psi_\omega y^{\mathbf{a}}\varepsilon e[{\mathbf{i}}]\Bigr)=0
\]
for coefficients $\lambda_{\mathbf{a},\omega,{\mathbf{i}}}\in \mathcal{Z}$ and project the above sum separately onto the idempotents $e_\Gamma({\mathbf{i}})$ and $e_{\Gamma'}({\mathbf{i}})$ for each ${\mathbf{i}}\in I^\alpha$;
we thus obtain sums of basis vectors of the form in Theorem \ref{qHbasis} which are linearly independent.
\end{proof}

We can obtain a $C_2$-graded Clifford decomposition for direct sums of blocks of quiver Hecke algebras using the element $\varepsilon$ defined in \eqref{varepdef}

\begin{Proposition}\label{clifforddecomp}
Let $\Gamma$ be a simply-laced quiver and let $2$ be invertible in $\mathcal{Z}$
 Then we have the $C_2$-graded Clifford decomposition
\[
\bigoplus_{\beta\in[\alpha]} \mathcal{R}_\beta \cong \mathcal{R}_{\alpha}(\Gamma/\Gamma')^\mathtt{sgn} \oplus \varepsilon \mathcal{R}_{\alpha}(\Gamma/\Gamma')^\mathtt{sgn}.
\]
\end{Proposition}

\begin{proof}
To demonstrate the Clifford decomposition, we check the requirements in Definition \ref{clifforddef}. Condition (i) follows easily; since $\varepsilon^\mathtt{sgn}=-\varepsilon$, if $x\in \mathcal{R}_n^\mathtt{sgn}$ and $y=\varepsilon z\in \varepsilon \mathcal{R}_n^\mathtt{sgn}$ then $(xy)^\mathtt{sgn}=-xy$ giving the required multiplicative property. Conditions (ii) and (iv) follow by definition, so it remains to demonstrate the direct sum decomposition.
Since any $x\in\mathcal{R}_\alpha\oplus\mathcal{R}_{\alpha'}$ may be written as $x=\sum_{{\mathbf{i}}\in I^\alpha\cup I^{\alpha'}}xe_\Gamma({\mathbf{i}})=\sum_{{\mathbf{i}}\in I^\alpha}x\Bigl[e_\Gamma({\mathbf{i}})+e_\Gamma(\tau({\mathbf{i}}))\Bigr]$, and since $2$ is invertible in $\mathcal{Z}$, we can write
\[
x=\frac12\sum_{{\mathbf{i}}\in I^\alpha}(x+x^\mathtt{sgn})\Bigl[e_\Gamma({\mathbf{i}})+e_\Gamma(\tau({\mathbf{i}}))\Bigr]+ \frac12\varepsilon\sum_{{\mathbf{i}}\in I^\alpha}(x-x^\mathtt{sgn})\Bigl[e_\Gamma({\mathbf{i}})+e_\Gamma(\tau({\mathbf{i}}))\Bigr]
\]
which gives the required decomposition
\end{proof}

\begin{Remark}\label{clifffailrk}$\;$
The reader may have noticed the dependence of our element $\varepsilon$ on choices of equivalence class representatives for each class $[{\mathbf{i}}]\in I^n_\tau$. Indeed, there are many choices of Clifford element which give rise to different Clifford decompositions, much in the way that different choices of coset representatives give rise to different Clifford decompositions of representations of finite groups with respect to normal subgroups \cite{CRM}.
\end{Remark}

\begin{Example}
We finish this chapter with an example of our results when $\Gamma=\Gamma_e$. First suppose $n=1$ and that $2<e<\infty$. Set $I=\mathbb{Z}/e\mathbb{Z}$ and notice that in this case our quiver Hecke algebra $\mathcal{R}_1(\Gamma_e,\mathcal{Z})$ has a particularly simple presentation:
\begin{align*}
\mathcal{R}_1(\Gamma_e)&=\Bigl\langle y,e(i)\;\Bigr|\;\Bigl. i\in I,\; ye(i)=e(i)y, \;\sum_{i\in I}e(i)=1, \;e(i)e(j)=\delta_{ij}e(i)\Bigr\rangle\\
&\cong \bigoplus_{i\in I}\mathcal{Z}[y]e(i).
\end{align*}
where $\deg y =2$ and $\deg e(i)=0$. This has the obvious homogeneous basis
\[
\{y^ke(i)\mid k\geq 0, i\in I\}.
\]

Suppose now that $e=3$ and that $\mathrm{char}(\mathcal{Z})\neq 2$; then it is not too hard to see that the following is a homogeneous basis for the fixed-point subalgebra $\mathcal{R}_1(\Gamma_e)^\mathtt{sgn}$, where we have grouped basis vectors by degree:
\begin{alignat*}{2}
\deg 0:&\quad 2e(0), \;\;e(1)+e(2)\\
\deg 2:&\quad y\Bigl(e(1)-e(2)\Bigr)\\
\deg 4:&\quad 2y^2 e(0), \;\;y^2\Bigl(e(1)+e(2)\Bigr)\\
\deg 6:&\quad y^3\Bigl(e(1)-e(2)\Bigr)\\
&\quad\qquad \vdots
\end{alignat*}
where $y=y_1$. For general $e>2$, we have the homogeneous basis
\[
\left\{y^k \Bigl(e(i)+e(-i)\Bigr)\mid k \;\mbox{even}\right\}\cup\left\{y^k\Bigl(e(i)-e(-i)\Bigr)\mid k\;\mbox{odd}, \; i\neq 0\right\}
\]
for $\mathcal{R}_n(\Gamma_e)^\mathtt{sgn}$. Note that
\begin{align*}
\dim \Bigl([\mathcal{R}_n(\Gamma_e)^\mathtt{sgn}]_{2k}&\oplus [\mathcal{R}_n(\Gamma_e)^\mathtt{sgn}]_{2(k+1)}\Bigr)\\
&\qquad =\frac12\dim \Bigl([\mathcal{R}_n(\Gamma_e)]_{2k}\oplus [\mathcal{R}_n(\Gamma_e)]_{2(k+1)}\Bigr)
\end{align*}
for all $k>0$.
\end{Example}

\section{Generators and relations for alternating quiver Hecke algebras}

Using the $C_2$-graded Clifford decomposition from Theorem \ref{aqHbasis}, we can give a presentation for the alternating quiver Hecke algebra $\mathcal{R}_n(\Gamma/\Gamma')^\mathtt{sgn}$ by homogeneous generators and relations, akin to the Khovanov-Lauda presentation for the quiver Hecke algebras (Definition \ref{D:klrdef}). The following elements will play the role of the generators from Definition \ref{D:klrdef} for alternating quiver Hecke algebras

\begin{Definition}[Generators for alternating quiver Hecke algebras]\label{aqHagens}
Let $\Gamma$ be a simply-laced quiver and $\mathcal{R}_n(\Gamma/\Gamma')^\mathtt{sgn}$ be an alternating quiver Hecke algebras. For $1\leq r<n$, define $\Psi_r=\psi_r\varepsilon\in \mathcal{R}_n(\Gamma/\Gamma')^\mathtt{sgn}$, and for $1\leq r\leq n$, define $\mathcal{Y}_r=y_r\varepsilon\in\mathcal{R}_n(\Gamma/\Gamma')^\mathtt{sgn}$. Finally, for $[{\mathbf{i}}]\in I^n_\tau$ recall the definition of $e[{\mathbf{i}}]\in\mathcal{R}_n(\Gamma/\Gamma')^\mathtt{sgn}$ from \eqref{esquaredef}.
\end{Definition}
\newcommand{\varepsilon}{\varepsilon}

As a corollary to Theorem \ref{aqHbasis} we see that the elements defined above generate the alternating quiver Hecke algebras.

\begin{Corollary}\label{aqHagenscor}
Let $\Gamma$ be a simply-laced quiver, let $n>1$ and $e>2$ and suppose 2 is invertible in $\mathcal{Z}$. For $[\alpha]\in (Q_\Gamma^+)_\tau$, the alternating quiver Hecke algebra $\mathcal{R}_{\alpha}(\Gamma/\Gamma')^\mathtt{sgn}$ is generated by the collection of elements
\[
\{\Psi_1,\Psi_2,\ldots,\Psi_{n-1}\}\cup\{\mathcal{Y}_1,\mathcal{Y}_2,\ldots,\mathcal{Y}_n\}\cup\{e[{\mathbf{i}}]\mid [{\mathbf{i}}]\in I^{[\alpha]}_\tau\}.
\]
\end{Corollary}
\begin{proof}
Suppose $[\alpha]\in (Q_\Gamma^+)_\tau$. We proceed to write each basis vector from Theorem \ref{aqHbasis} in terms of the proposed generators; since the $y_r$ commute with the $e_\Gamma({\mathbf{i}})$ for all admissible $r$ and ${\mathbf{i}}$ by the relations in Definition \ref{D:klrdef}, and since $\varepsilon^2=1$,
\[
y_1^{a_1}\cdots y_n^{a_n}=\left\{\begin{array}{ll} \mathcal{Y}_1^{a_1}\cdots \mathcal{Y}_n^{a_n},&\mbox{if $\left|\mathbf{a}\right|\equiv 0\;\mbox{mod}\;2$}\\ \varepsilon\mathcal{Y}_1^{a_1}\cdots \mathcal{Y}_n^{a_n},&\mbox{if $\left|\mathbf{a}\right|\equiv 1\;\mbox{mod}\;2$}\end{array}\right.
\]
Moreover, since $\psi_\omega e_\Gamma({\mathbf{i}})=e_\Gamma(\omega\cdot{\mathbf{i}})\psi_\omega$ by the relations in Definition \ref{D:klrdef},
\[
\psi_\omega= \left\{\begin{array}{ll} \Psi_\omega,&\mbox{if $\ell(\omega)\equiv 0\;\mbox{mod}\;2$}\\ \Psi_\omega\varepsilon,&\mbox{if $\ell(\omega)\equiv 1\;\mbox{mod}\;2$}\end{array}\right.
\]
There are four cases to consider; we just give two as illustration of the method of proof. If $\left|\mathbf{a}\right|\equiv 0\;\mbox{mod}\;2$ and $\ell(\omega)\equiv 0\;\mbox{mod}\;2$, then the basis vector $\psi_\omega y_1^{a_1}\cdots y_n^{a_n}e[{\mathbf{i}}]$ is equal to $\Psi_{i_1}\cdots\Psi_{i_d}\mathcal{Y}_1^{a_1}\cdots\mathcal{Y}_n^{a_n} e[{\mathbf{i}}]$ where $\omega=s_{i_1}\cdots s_{i_d}$. If $\left|\mathbf{a}\right|\equiv 1\;\mbox{mod}\;2$ and $\ell(\omega)\equiv 0\;\mbox{mod}\;2$ then the basis vector
\begin{align*}
\psi_\omega y_1^{a_1}\cdots y_n^{a_n}[e_\Gamma({\mathbf{i}})-e_\Gamma(\tau({\mathbf{i}}))]&= \psi_\omega y_1^{a_1}\cdots y_n^{a_n}\varepsilon e[{\mathbf{i}}]\\
&= \Psi_{i_1}\cdots\Psi_{i_d} \varepsilon \mathcal{Y}_1^{a_1}\cdots \mathcal{Y}_n^{a_n}\varepsilon e[{\mathbf{i}}]\\
&= \Psi_{i_1}\cdots\Psi_{i_d} \mathcal{Y}_1^{a_1}\cdots \mathcal{Y}_n^{a_n} e[{\mathbf{i}}]
\end{align*}
since $\varepsilon$ commutes with the $\mathcal{Y}_r$ and squares to 1.
\end{proof}

Our goal now is to obtain a set of {\em relations} for the generators from Corollary \ref{aqHagenscor}. Let us start by defining a new abstract algebra with a slightly different presentation and the additional structure of a $\mathbb{Z}\times C_2$-grading; as in \S5.1, let us realise $C_2$ as the sign group $\{+,-\}$ with usual multiplication of signs

\begin{Definition}\label{twisteddef}
Let $n\geq 0$. The {\em signed quiver Hecke algebra} of type $\Gamma$ and corresponding to $[\alpha]\in (Q^+_{\Gamma})_\tau$ is the unital associative $\mathcal{Z}$-algebra $\mathcal{R}_{[\alpha]}(\Gamma)'=\mathcal{R}_{[\alpha]}'(\Gamma,\mathcal{Z})$ with generators
\[
\{\psi_1',\psi_2',\ldots,\psi_{n-1}'\}\cup\{y_1',y_2',\ldots,y_n'\}\cup\Bigl\{\varepsilon_a({\mathbf{i}})\mid {\mathbf{i}}\in \bigcup_{\beta\in [\alpha]}I^\beta,\; a\in C_2\Bigr\}
\]
subject to the relations
\begin{align*}
\varepsilon_a({\mathbf{i}})\varepsilon_b({\mathbf{j}})&= \delta_{{\mathbf{i}}{\mathbf{j}}}\varepsilon_{ab}({\mathbf{i}}),\qquad \sum_{{\mathbf{i}}\in I^{[\alpha]}}\varepsilon_+({\mathbf{i}})= 1\\
\varepsilon_a({\mathbf{i}})&= a\varepsilon_a(\tau({\mathbf{i}}))\\
y_r'\varepsilon_a({\mathbf{i}})&= \varepsilon_a({\mathbf{i}})y_r'\\
\psi_r'\varepsilon_a({\mathbf{i}})&= \varepsilon_a(s_r\cdot{\mathbf{i}})\psi_r'\\
y_r'y_s'\varepsilon_-({\mathbf{i}})&= y_s'y_r'\varepsilon_-({\mathbf{i}})\\
\psi_r'y_s'\varepsilon_-({\mathbf{i}})&= y_s'\psi_r'\varepsilon_-({\mathbf{i}})\quad\mbox{if $s\neq r,r+1$}\\
\psi_r'\psi_s'\varepsilon_-({\mathbf{i}})&= \psi_s'\psi_r'\varepsilon_-({\mathbf{i}})\quad\mbox{if $\left|r-s\right|>1$}\\
\psi_r'y_{r+1}'\varepsilon_a({\mathbf{i}})&= \left\{\begin{array}{ll}(y_r'\psi_r'+1)\varepsilon_a({\mathbf{i}}),&\mbox{if $i_r=i_{r+1}$}\\
y_r'\psi_r'\varepsilon_a({\mathbf{i}}),&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.\\
y_{r+1}'\psi_r'\varepsilon_a({\mathbf{i}})&= \left\{\begin{array}{ll}(\psi_r'y_r'+1)\varepsilon_a({\mathbf{i}}),&\mbox{if $i_r=i_{r+1}$}\\\psi_r'y_r'\varepsilon_a({\mathbf{i}}),&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.\\
(\psi_r')^2\varepsilon_-({\mathbf{i}})&= \left\{\begin{array}{ll} 0,&\mbox{if $i_r=i_{r+1}$}\\
(y_r'-y_{r+1}')\varepsilon_+({\mathbf{i}}),&\mbox{if $i_r\to i_{r+1}$}\\
(y_{r+1}'-y_r')\varepsilon_+({\mathbf{i}}),&\mbox{if $i_r\leftarrow i_{r+1}$}\\
\varepsilon_-({\mathbf{i}}),&\mbox{otherwise}\end{array}\right.\\
\psi_r'\psi_{r+1}'\psi_r'\varepsilon_+({\mathbf{i}})&=\left\{\begin{array}{ll} \psi_{r+1}'\psi_r'\psi_{r+1}'\varepsilon_+({\mathbf{i}})+\varepsilon_-({\mathbf{i}}),&\mbox{if $i_{r+2}=i_r\to i_{r+1}$}\\
\psi_{r+1}'\psi_r'\psi_{r+1}'\varepsilon_+({\mathbf{i}})-\varepsilon_-({\mathbf{i}}),&\mbox{if $i_{r+2}=i_r\leftarrow i_{r+1}$}\\
\psi_{r+1}'\psi_r'\psi_{r+1}'\varepsilon_+({\mathbf{i}}),&\mbox{otherwise}\end{array}\right.
\end{align*}
for ${\mathbf{i}},{\mathbf{j}}\in I^\alpha$, $a,b\in C_2$ and all admissible $r$ and $s$.
\end{Definition}

\begin{Remark}$\;$
\begin{itemize}
\item[(i)]  As usual, the algebra $\mathcal{R}_{[\alpha]}'$ is a block of the full signed quiver Hecke algebra $\mathcal{R}_n'$, which decomposes as
\[
\mathcal{R}_n(\Gamma)'=\mathcal{R}_n'(\Gamma,\mathcal{Z})=\bigoplus_{[\alpha]\in (Q^+_{\Gamma})_\tau}\mathcal{R}_{[\alpha]}'
\]
and $\mathcal{R}_\alpha'=\varepsilon_{\alpha}^+\mathcal{R}_n'$ where $\varepsilon_{\alpha}^+=\sum_{{\mathbf{i}}\in I^\alpha}\varepsilon_+({\mathbf{i}})$.
\item[(ii)] The generators in Definition \ref{twisteddef} are somewhat superfluous: for example we can use the relations to see that $\varepsilon_a({\mathbf{i}})=0$ whenever ${\mathbf{i}}=\tau({\mathbf{i}})$. These extra generators do however mean we can more easily compare the algebra $\mathcal{R}'_{[\alpha]}$ with the quiver Hecke algebra, which we are about to do.
\end{itemize}

\end{Remark}

The proof of the following lemma requires nothing more than an inspection of the relations in Definition \ref{twisteddef} which we leave to the reader (see \S2.1 for general remarks on algebras graded by a finite group $G$).

\begin{Lemma}\label{z2grading}
The relations in Definition \ref{twisteddef} are homogeneous with respect to the degree function $\deg_2:\mathcal{R}_n'\to (\mathbb{Z}\times C_2)$ given by
\[
\deg_2  \varepsilon_a({\mathbf{i}})= (0,a),\qquad \deg_2 y_r'=(2,-),\qquad \begin{array}{l}\deg_2\psi_r'\varepsilon_+({\mathbf{i}})= (-c_{i_ri_{r+1}},-)\\\deg_2\psi_r'\varepsilon_-({\mathbf{i}})= (-c_{i_ri_{r+1}},+)\end{array}
\]
for all ${\mathbf{i}}\in I^n$, $a\in C_2$ and $1\leq r\leq n$. In particular, $\mathcal{R}'_n$ is a $(\mathbb{Z}\times C_2)$-graded $\mathcal{Z}$-algebra.
\end{Lemma}

\begin{Remark}
By forgetting the $C_2$-grading (but {\em not} the $\mathbb{Z}$-grading) on $\mathcal{R}'_n$ we obtain a $\mathbb{Z}$-graded algebra.
\end{Remark}

\begin{Proposition}
Let $n> 0$ and suppose $e> 2$. If $2$ is invertible in $\mathcal{Z}$, then for $[\alpha]\in (Q^+_\Gamma)_\tau$,
\[
\mathcal{R}'_{[\alpha]}(\Gamma,\mathcal{Z})\cong \bigoplus_{\beta\in[\alpha]}\mathcal{R}_{\beta}(\Gamma,\mathcal{Z})
\]
as $\mathbb{Z}$-graded $\mathcal{Z}$-algebras.
\end{Proposition}
\begin{proof}
Define a map $\vartheta:\mathcal{R}'_{[\alpha]}(\Gamma,\mathcal{Z})\to \bigoplus_{\beta\in[\alpha]}\mathcal{R}_{\beta}(\Gamma,\mathcal{Z})$ on generators by
\begin{align*}
\vartheta(y_r')&= y_r\\
\vartheta(\psi_s')&= \psi_s\\
\vartheta(\varepsilon_a({\mathbf{i}}))&= \left\{\begin{array}{ll}e({\mathbf{i}})+e(\tau({\mathbf{i}})),&\mbox{if $a=+$}\\e({\mathbf{i}})-e(\tau({\mathbf{i}})),&\mbox{if $a=-$}\end{array}\right.
\end{align*}
for $1\leq r\leq n$ and $1\leq s<n$ and ${\mathbf{i}}\in I^\alpha$. We must check $\vartheta$ is an algebra homomorphism of degree zero; this amounts to the largely tedious and straightforward calculation of checking it preserves the relations. We check two relations so the reader can obtain a taste for how they are done. For example, let ${\mathbf{i}},{\mathbf{j}}\in I^\alpha$. Then we have
\begin{align*}
\vartheta(\varepsilon_a({\mathbf{i}})\varepsilon_b({\mathbf{j}}))&= \vartheta(\delta_{{\mathbf{i}}{\mathbf{j}}}\varepsilon_{ab}({\mathbf{i}}))\\
&= \left\{\begin{array}{ll}\delta_{{\mathbf{i}}{\mathbf{j}}}(e({\mathbf{i}})+e(\tau({\mathbf{i}}))),&\mbox{if $a=b$}\\\delta_{{\mathbf{i}}{\mathbf{j}}}(e({\mathbf{i}})-e(\tau({\mathbf{i}}))),&\mbox{if $a\neq b$}\end{array}\right.\\
&= \left\{\begin{array}{ll}\delta_{{\mathbf{i}}{\mathbf{j}}}(e({\mathbf{i}})+e(\tau({\mathbf{i}}))),&\mbox{if $a=b$}\\\delta_{{\mathbf{i}}{\mathbf{j}}}(e({\mathbf{i}})-e(\tau({\mathbf{i}}))),&\mbox{if $a\neq b$}\end{array}\right.\\
&= \vartheta(\varepsilon_a({\mathbf{i}}))\vartheta(\varepsilon_b({\mathbf{j}})).
\end{align*}
Now suppose that ${\mathbf{i}}\in I^\alpha$ is such that $i_{r+2}=i_r\to i_{r+1}$. Then
\begin{align*}
\vartheta(\psi_r'\psi_{r+1}'\psi_r'\varepsilon_+({\mathbf{i}}))&= \vartheta(\psi_{r+1}'\psi_r'\psi_{r+1}'\varepsilon_+({\mathbf{i}})+\varepsilon_-({\mathbf{i}}))\\
&= \psi_{r+1}\psi_r\psi_{r+1}[e({\mathbf{i}})+e(\tau({\mathbf{i}}))]+e({\mathbf{i}})-e(\tau({\mathbf{i}}))\\
&= (\psi_{r+1}\psi_r\psi_{r+1}+1)e({\mathbf{i}})+(\psi_{r+1}\psi_r\psi_{r+1}-1)e(\tau({\mathbf{i}}))\\
&= \psi_r\psi_{r+1}\psi_r(e({\mathbf{i}})+e(\tau({\mathbf{i}})))\\
&= \vartheta(\psi_r')\vartheta(\psi_{r+1}')\vartheta(\psi_r')\vartheta(\varepsilon_+({\mathbf{i}}))
\end{align*}
as required. We leave the remaining checks to the reader; this amounts to proving $\vartheta$ is surjective. Similarly, define a map $\varsigma:\bigoplus_{\beta\in[\alpha]}\mathcal{R}_\beta\to \mathcal{R}_{[\alpha]}'$ on generators by
\begin{align*}
\varsigma(y_r)&= y_r'\\
\varsigma(\psi_s)&= \psi_s'\\
\varsigma(e({\mathbf{i}}))&= \frac12\left(\varepsilon_+({\mathbf{i}})+\varepsilon_-({\mathbf{i}})\right)
\end{align*}
for $1\leq r\leq n$, $1\leq s< n$ and ${\mathbf{i}}\in\bigcup_{\beta\in [\alpha]}I^\beta$. We must check that $\varsigma$ extends to an algebra homomorphism, again by checking it preserves all relations: note that since $\varepsilon_a({\mathbf{i}})\varepsilon_b({\mathbf{i}})=\varepsilon_{a+b}({\mathbf{i}})$ for all ${\mathbf{i}}\in I^n$ and $a,b\in\mathbb{Z}_2$, we can multiply all the relations in Definition \ref{twisteddef} by $\varepsilon_-({\mathbf{i}})$ to obtain a list of additional relations which also hold in $\mathcal{R}_{[\alpha]}'$, obtaining relations like
\[
\psi_r'y_s'\varepsilon_+({\mathbf{i}})=y_s'\psi_r'\varepsilon_+({\mathbf{i}}).
\]
These, together with the original list of relations, allow one to check that $\varsigma$ respects all of the relations. For example, if we compute
\[
\varsigma(y_r\psi_se({\mathbf{i}}))=\frac12\Bigl(y_r'\psi_s'\varepsilon_+({\mathbf{i}})+y_r'\psi_s'\varepsilon_-({\mathbf{i}})\Bigr)
\]
we can see that both terms on the right-hand side are indeed relations in $\mathcal{R}'_{[\alpha]}$ (with the first term being a relation from Definition \ref{twisteddef} multiplied on the right by $\varepsilon_-({\mathbf{i}})$).
Since $\varsigma$ is also homogeneous of degree zero and since we clearly have $\vartheta\circ\varsigma=\mathrm{id}$ and $\varsigma\circ\vartheta=\mathrm{id}$, this establishes the required isomorphism of $\mathbb{Z}$-graded algebras.
\end{proof}

Using the $C_2$-grading, we can write the decomposition of $\mathcal{R}_n'$ into odd and even parts afforded by its $C_2$ grading as $\mathcal{R}_n'=(\mathcal{R}_n')_+\oplus (\mathcal{R}_n')_-$. This, combined with Theorem \ref{aqHbasis}, gives the following corollary regarding the alternating quiver Hecke algebra.

\begin{Corollary}\label{Z2decomp}
Let $n\geq 0$, let $\Gamma$ be a simply-laced quiver, and let $2$ be invertible in $\mathcal{Z}$. Then $(\mathcal{R}_n(\Gamma)')_+\cong \mathcal{R}_n(\Gamma/\Gamma')^\mathtt{sgn}$.
\end{Corollary}

\renewcommand{{\mathbb{Y}}}{\mathcal{Y}}

\begin{Theorem}[Generators and relations for alternating quiver Hecke algebras]\label{aqHapres}
Let $n>1$, let $(\Gamma,\tau)$ be a reversible quiver and suppose 2 is invertible in $\mathcal{Z}$.
Let $[\alpha]\in Q^+_\tau$. Then the alternating quiver Hecke algebra $\mathcal{R}_\alpha(\Gamma/\Gamma')^\mathtt{sgn}$ is generated by the elements
\[
\{\Psi_1,\Psi_2,\ldots,\Psi_{n-1}\}\cup\{{\mathbb{Y}}_1,{\mathbb{Y}}_2,\ldots,{\mathbb{Y}}_n\}\cup \{e[{\mathbf{i}}]\mid [{\mathbf{i}}]\in I^{[\alpha]}_\tau\}
\]
subject to the relations
\begin{align*}
e[{\mathbf{i}}]e[{\mathbf{j}}]&= \delta_{[{\mathbf{i}}][{\mathbf{j}}]}e[{\mathbf{i}}],\qquad\qquad \sum_{[{\mathbf{i}}]\in I^{[\alpha]}_\tau}e[{\mathbf{i}}]= 1\\
{\mathbb{Y}}_r e[{\mathbf{i}}]&= e[{\mathbf{i}}]{\mathbb{Y}}_r\\
\Psi_r e[{\mathbf{i}}]&=e[s_r\cdot{\mathbf{i}}]\Psi_r\\%\left\{\begin{array}{ll},&\mbox{if $s_r\cdot{\mathbf{i}}\in I^n_+$}\\ -e^+(-s_r\cdot{\mathbf{i}})\Psi_r,&\mbox{if $s_r\cdot{\mathbf{i}}\in I^n_-$}\end{array}\right.\\
{\mathbb{Y}}_r{\mathbb{Y}}_s&= {\mathbb{Y}}_s{\mathbb{Y}}_r\\
\Psi_r{\mathbb{Y}}_se[{\mathbf{i}}]&={\mathbb{Y}}_s\Psi_re[{\mathbf{i}}]\quad\mbox{if $s\neq r,r+1$}\\
\Psi_r\Psi_se[{\mathbf{i}}]&= \Psi_s\Psi_re[{\mathbf{i}}],\quad\mbox{if $\left|r-s\right|>1$}\\
\Psi_r{\mathbb{Y}}_{r+1}e[{\mathbf{i}}]&= \left\{\begin{array}{ll} ({\mathbb{Y}}_r\Psi_r + 1)e[{\mathbf{i}}],&\mbox{if $i_r=i_{r+1}$}\\
{\mathbb{Y}}_r\Psi_r e[{\mathbf{i}}],&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.\\
{\mathbb{Y}}_{r+1}\Psi_re[{\mathbf{i}}]&= \left\{\begin{array}{ll} (\Psi_r{\mathbb{Y}}_r+1)e[{\mathbf{i}}],&\mbox{if $i_r=i_{r+1}$}\\
\Psi_r{\mathbb{Y}}_re[{\mathbf{i}}],&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.
\end{align*}
\begin{align*}
\Psi_r^2e[{\mathbf{i}}]&= \left\{\begin{array}{ll}
0,&\mbox{if $i_r=i_{r+1}$}\\
({\mathbb{Y}}_r-{\mathbb{Y}}_{r+1})e[{\mathbf{i}}],&\mbox{if $i_r\to i_{r+1}$}\\
({\mathbb{Y}}_{r+1}-{\mathbb{Y}}_r)e[{\mathbf{i}}],&\mbox{if $i_r\leftarrow i_{r+1}$}\\
e[{\mathbf{i}}],&\mbox{otherwise}\end{array}\right.\\
\Psi_r\Psi_{r+1}\Psi_re[{\mathbf{i}}]&= \left\{\begin{array}{l} (\Psi_{r+1}\Psi_r\Psi_{r+1}-1)e[{\mathbf{i}}],\\
\qquad\qquad\qquad\quad\mbox{if $i_r=i_{r+2}\to i_{r+1}$}\\
(\Psi_{r+1}\Psi_r\Psi_{r+1}+1)e[{\mathbf{i}}],\\
\qquad\qquad\qquad\qquad\mbox{if $i_r=i_{r+2}\leftarrow i_{r+1}$}\\
\Psi_{r+1}\Psi_r\Psi_{r+1}e[{\mathbf{i}}],\\
\qquad\qquad\qquad\quad\mbox{otherwise}\end{array}\right.
\end{align*}
for all $[{\mathbf{i}}]\in I^{[\alpha]}_\tau$ and all admissible $r$ and $s$.
Moreover, $\mathcal{R}_\alpha(\Gamma/\Gamma')^\mathtt{sgn}$ is $\mathbb{Z}$-graded with degree function
\[
\deg\Psi_re[{\mathbf{i}}]=-a_{i_r i_{r+1}},\quad \deg{\mathbb{Y}}_s=2,\quad \deg e[{\mathbf{i}}]=0.
\]
for all $1\leq r<n$, $1\leq s\leq n$ and $[{\mathbf{i}}]\in I^{[\alpha]}_\tau$.
\end{Theorem}

\begin{proof}
First we note that all the requirements on the residue sequences ${\mathbf{i}}$ in fact depend only on the {\em equivalence class} of the sequence because of the symmetry of the Cartan matrix in \eqref{cartanmatrixdef}. By Corollary \ref{Z2decomp}, it is enough to prove that the abstract algebra $A_{[\alpha]}$ defined in the statement of the theorem is isomorphic to $(\mathcal{R}_{[\alpha]}')_+$. Define a map $\varrho:A_{[\alpha]}\to (\mathcal{R}_{[\alpha]}')_+$ by
\[
e[{\mathbf{i}}]\mapsto \varepsilon_+({\mathbf{i}}),\quad \mathcal{Y}_r\mapsto y_r'\varepsilon_-({\mathbf{i}}),\quad \Psi_s\mapsto \psi_s'\varepsilon_-({\mathbf{i}}),
\]
for all $[{\mathbf{i}}]\in I^{[\alpha]}_\tau$, $1\leq r\leq n$ and $1\leq s<n$. It is a straightforward check that all the relations of $A_{[\alpha]}$ are satisfied in $(\mathcal{R}_{[\alpha]}')_+$, so $\varrho$ determines a well-defined homogeneous algebra homomorphism of degree zero.

By definition, $(\mathcal{R}_{[\alpha]}')_+$ is generated by even words in the generators of $\mathcal{R}_{[\alpha]}'$. However the only even {\em generators} of $\mathcal{R}_{[\alpha]}'$ are the idempotents $\varepsilon_+({\mathbf{i}})$ for ${\mathbf{i}}\in I^\alpha\cup I^{\alpha'}$, so $(\mathcal{R}_{[\alpha]}')_+$ is generated by these idempotents together with all words of even length in the {\em odd} generators of $\mathcal{R}_{[\alpha]}'$. It is now easy to see that $(\mathcal{R}_{[\alpha]}')_+$ is generated by the images of the generators of $A_{[\alpha]}$ under $\varrho$ and so $\varrho$ is surjective.

The algebra $\mathcal{R}_{[\alpha]}'$ is defined by generators and relations, so $(\mathcal{R}_{[\alpha]}')_+$ is the subalgebra of $\mathcal{R}_{[\alpha]}'$ generated by the even words in the generators of $\mathcal{R}_{[\alpha]}'$ modulo the even part of the relation ideal defining $\mathcal{R}_{[\alpha]}'$, which is the set of all linear combinations of arbitrary products of even relations multiplied by even products of odd relations.
However the only even relations in $\mathcal{R}_{[\alpha]}'$ are given by idempotents and commutation relations.

Therefore, the even part of the relation ideal for $\mathcal{R}_{[\alpha]}'$ is generated by the even relations in $\mathcal{R}_{[\alpha]}'$ together with all products of the odd relations in $\mathcal{R}_{[\alpha]}'$.
It follows that all the remaining relations are generated by even products of odd relations, together with odd relations multiplied by $\varepsilon$: in this way we obtain the complete set of relations for $(\mathcal{R}_{[\alpha]}')_+$.
One checks these are precisely the relations written above for $A_{[\alpha]}$; for example multiplying the relation
\[
y_{r+1}'\psi_r'\varepsilon_-({\mathbf{i}})=\left\{\begin{array}{ll}(\psi_r'y_r'+1)\varepsilon_-({\mathbf{i}}),&\mbox{if $i_r=i_{r+1}$}\\ \psi_r'y_r'\varepsilon_-({\mathbf{i}}),&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.
\]
by $\varepsilon_-({\mathbf{i}})$ and using the idempotent relations in $\mathcal{R}_{[\alpha]}'$ gives the relation
\[
y_{r+1}'\varepsilon_-({\mathbf{i}})\psi_r'\varepsilon_-({\mathbf{i}})\varepsilon_+({\mathbf{i}})=\left\{\begin{array}{ll}(\psi_r'\varepsilon_-({\mathbf{i}})y_{r+1}'\varepsilon_-({\mathbf{i}})+1)\varepsilon_+({\mathbf{i}}),&\mbox{if $i_r=i_{r+1}$}\\ \psi_r'\varepsilon_-({\mathbf{i}}) y_r'\varepsilon_-({\mathbf{i}})\varepsilon_+({\mathbf{i}}),&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.
\]
which is precisely the image of the relation
\[
\mathcal{Y}_{r+1}\Psi_re[{\mathbf{i}}]=\left\{\begin{array}{ll}(\Psi_r\mathcal{Y}_r+1)e[{\mathbf{i}}],&\mbox{if $i_r=i_{r+1}$}\\ \Psi_r\mathcal{Y}_re[{\mathbf{i}}],&\mbox{if $i_r\neq i_{r+1}$}\end{array}\right.
\]
under $\varrho$. Continuing in this way we see $\varrho$ is an isomorphism.

Finally, we note that the given degree function is well-defined; since the Cartan matrix $(c_{ij})_{i,j\in I}$ is symmetric, the entries only depend on equivalence classes of residue sequences under $\tau$.

\end{proof}

\begin{Remark}$\;$

\begin{itemize}
\item[(i)] Since we have given an isomorphism between the alternating quiver Hecke algebra and the even part of the abstract algebra $\mathcal{R}_n'$, for which we gave an abstract presentation by generators and relations, we see that the generators from Definition \ref{aqHagens} do not depend on the choice of equivalence class representatives ${\mathbf{i}}_\tau$ (see p\pageref{eiplusdef}).
\item[(ii)] In \cite[\S5.4]{BoysThesis} and \cite{BM}, cyclotomic quotients of the algebras from this paper are studied. In particular, \cite[Theorem B]{BM} gives a presentation for a cyclotomic quotient of the alternating quiver Hecke algebras by generators and relations very similar to those in this paper, in the special case when $\Gamma=\Gamma_e$ and the cyclotomic relation is simple. In the cyclotomic case when the cyclotomic parameter $\Lambda=\Lambda_0$, it is shown in \cite[Chapter 6]{BoysThesis} that alternating cyclotomic quiver Hecke algebras are isomorphic to alternating Hecke algebras in the sense of Mitsuhashi \cite{Mitsuhashi:A}, giving an analogue of Brundan and Kleshchev's result \cite{BK:GradedKL}.
\end{itemize}
\end{Remark}

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redpajama_set_name: RedPajamaArXiv


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