text:
R.H.C. Davis
A History of Medieval Europe
From Constantine to Saint Louis
Author: R.H.C. Davis
R.C. Davis provided the classic account of the European medieval world; equipping generations of undergraduate and 'A' level students with sufficient grasp of the period to debate diverse historical perspectives and reputations. His book has been important grounding for both modernists required to take a course in medieval history, and those who seek to specialise in the medieval period. In updating this classic work to a third edition, the additional author now enables students to see history in action; the diverse viewpoints and important research that has been undertaken since Davis' second edition, and progressed historical understanding. Each of Davis original chapters now concludes with a 'new directions and developments' section by Professor RI Moore, Emeritus of Newcastle University. A key work updated in a method that both enhances subject understanding and sets important research in its wider context. A vital resource, now up-to-date for generations of historians to come.
Ralph Henry Carless Davis
Author: Ralph Henry Carless Davis
To many of us the history of Europe from the age of Constantine to the middle of the thirteenth century is dark and confused. In his successful book, which is based on background lectures to students, Professor Davis has concentrated on the most important topics, and largely on western European themes, to provide a well-lighted and comprehensible pattern. In Part I, the Dark Ages, he begins with the new capital of the Roman Empire and the impact of the barbarian invasions. Then follow studies of the Church and the Papacy and of the rise of Islam. After chapters on the rise and fall of the Frankish empire, the part ends with an economic survey of Europe at the end of the ninth century. The second half of the book begins with the Saxon Empire and has chapters on monasticism, the reform of the Papacy, the Crusades and the feudal monarchy in France. It concludes with accounts of St. Francis, Frederick II and Louis IX. One of the unusual features of Part II is the completion of each chapter with translations of original documents giving the reader not only entertainment but also a hint of the sources of the writing of history. Professor Davis has a clear style, an eye for the essential and an ear for the apt quotation. He has written an admirable introduction, at the level of first-year undergraduates, to a fascinating but difficult period of history. It is now issued in paperback for the first time.
Publisher: London ; Toronto : Longmans, Green, 1957, 1958 printing.
Covers European history from the proclamation of Constantine the Great as Caesar in 306 to the death of Louis IX in 1270.
William J. Duiker
Author: William J. Duiker
Publisher: Cengage Learning
Noted teachers and scholars William J. Duiker and Jackson J. Spielvogel present a balanced, highly readable overview of world history that explores common challenges and experiences that unite the human past and identify key global patterns over time. Tho
M. Omidsalar
Poetics and Politics of Iran's National Epic, the Sh?hn?meh
Author: M. Omidsalar
This book considers some of the Western interpretations of The Shahnameh - Iran's national epic, and argues that these interpretations are not only methodologically flawed, but are also more revealing of Western concerns and anxieties about Iran than they are about the Shahnameh.
Jackson J. Spielvogel
Western Civilization: Volume A: To 1500
Author: Jackson J. Spielvogel
Best-selling author Jackson Spielvogel helped over one million students learn about the present by exploring the past. Spielvogel's engaging, chronological narrative weaves the political, economic, social, religious, intellectual, cultural, and military aspects of history into a gripping story that is as memorable as it is instructive. WESTERN CIVILIZATION includes three to ten maps in each chapter and numerous excerpts from primary sources that enliven the past while introducing students to the source material of historical scholarship. Additionally, the text is lushly illustrated with photographs that add visual context. A variety of pedagogical tools, including features on relevant films and new end-of-chapter study aids, make this edition accessible to any learning style. Available in the following split options: WESTERN CIVILIZATION, Eighth Edition (Chapters 1-30), ISBN: 978-0-495-91324-5; Volume I: To 1715 (Chapters 1-16), ISBN: 978-1-111-34212-8; Volume II: Since 1500 (Chapters 13-30), ISBN: 978-1-111-34213-5; Volume A: To 1500 (Chapters 1-12), ISBN: 978-1-111-34214-2; Volume B: 1300 to 1815 (Chapters 11-19), ISBN: 978-1-111-34215-9; Volume C: Since 1789 (Chapters 19-30), ISBN: 978-1-111-34216-6; Title: Alternate Volume: Since 1300 (Chapters 11-30), ISBN: 978-1-111-34219-7. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Victoria Clark
The Far-Farers
A Journey from Viking Iceland to Crusader Jerusalem
Author: Victoria Clark
Just before the year 1000 a young Viking named Thorvald the Far-farer turned his back on the pagan gods of his fathers to preach the Christian gospel, travelling to Jerusalem, the golden heart of all medieval world maps. A thousand years later Victoria Clark retraces his epic voyage to discover how the dramatic events of Thorvald's Europe still resonate today. This is a compelling, highly acclaimed blend of history and travel, in the manner of William Dalrymple's bestselling From the Holy Mountain. 'Deeply engaging . . . The author's great sensitivity shines as brightly as ever it did in Why Angels Fall' Independent on Sunday 'She writes books whose ambition and impressiveness must leave most of her journalist friends ill with envy' Financial Times 'Entertaining, instructive and relevant' Sunday Times Book of the Week 'I read every word of it, and went back over some of the chapters and read them again for sheer pleasure. A triumph' John Cornwell, author of Hitler's Pope
Western Civilization: A Brief History
Put the world today into context by learning about the past through this brief, best-selling Western Civilization text that has helped thousands of students succeed in the course. Jack Spielvogel's engaging style of writing weaves the political, economic, social, religious, intellectual, cultural, and military aspects of history into a gripping story that is as memorable as it is instructive. You will also be exposed to primary source documents--actual historical documents that are the foundation for the historical analysis you read in the chapter. These documents include letters, poems, and songs through history--documents that enliven the past. Throughout the book there are also helpful tools to help you digest the reading, including outlines, focus questions, chronologies, numerous maps, and boldface key terms with definitions. Available with InfoTrac Student Collections http://gocengage.com/infotrac. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Kelly Robert DeVries
Medieval Military Technology, Second Edition
Author: Kelly Robert DeVries
Publisher: University of Toronto Press
First published in 1992, Medieval Military Technology has become the definitive book in its field, garnering much praise and a large readership. This thorough update of a classic book, regarded as both an excellent overview and an important piece of scholarship, includes fully revised content, new sections on the use of horses, handguns, incendiary weapons, and siege engines, and eighteen new illustrations. The four key organizing sections of the book still remain: arms and armor, artillery, fortifications, and warships. Throughout, the authors connect these technologies to broader themes and developments in medieval society as well as to current scholarly and curatorial controversies.
Fugitive Fiancee (Maitland Maternity)
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Nottingham, York & the North East
The 1970s Scrapbook
Y Vino Un Hombre de Dios El Cual Se Llam Eusebio
Building A Story Brand
Roman Coins and Their Values Volume 3
The Rebirths Of Tao
Lonely Planet Discover New Zealand
Cyfres Ysgwyd Cnoi a Dysgu: Teganau
Basher Science: The Complete Periodic Table
Beginners Bible for Toddlers
Memoirs of a Fruitcake
Complete Guitar Repair
Handbook of Thanatology
Database System Concepts (Irwin Computer Science)
Children, their World, their Education
Princess Mirror-Belle
Designing Costume for Stage and Screen
Son of Stone: A Stone Barrington Novel
Faces of Britain
Nate The Great And The Snowy Trail
Ghosts And The Japanese
One True Thing: A Novel
Action Man Annual
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Honda 250-350cc Twins 64-74
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Mindful Monkey, Happy Panda
The Little Book on How to be a Lady
Windows 10 for Seniors and Beginners
American Military Training Aircraft
Prez Vol. 1
Black Order
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Nutrition: Concepts and Controversies
Ella Diaries Super Fantastic Collection
Make Your Own Living Trust
Freeing Yourself Fro the Narcissist in Your Life
The Monster Who Lost His Mean
The Land of Laughs
Through an Indian's Looking Glass
The Wicked + The Divine: Book One
I am an Aspie Girl
Psalms Volume 1
Learning Law
Classic Starts (R): The Hunchback of Notre-Dame
Ein Treffen mit Louise L. Hay, 1 Videocassette
A Summer Seduction
Practical Self Sufficiency


Athens is the birthplace of European civilization, a city which inspired modern democracy and spawned three of the world's greatest ever philosophers Aristotle, Plato and Socrates. Inhabited for nearly 7,000 years, the city has been ravaged by interminable wars which have done more to destroy its former splendour than time and the forces of nature.
The city's history is inextricably woven into the ruined walls of its awe inspiring Acropolis one of the world's most famous monuments which attracts no fewer than three million visitors a year. The word acropolis comes from acro (meaning high point) and polis (meaning city). All major settlements throughout ancient Greece had their own acropolis which provided a defence against the threat of invasion.
Around 5000 BC Neolithic settlers were drawn to the Athens hilltop because of its steep slopes and the abundant supply of water from two springs. By 1400 BC a powerful city had developed on the Acropolis under the Mycenaean Civilisation which saw various kings inhabiting fortified palaces on hill tops which could be easily defended.
The 6th century BC saw the emergence of an early form of democracy when the relatively enlightened chief magistrate Solon introduced some radical reforms, including trial by jury. In 510 BC, after the city managed to rid itself of the tyrant Hippias, the Oracle at Delphi decreed that the Acropolis should in future remain the province of the gods and should never more be occupied by humans. The city was about to enter its glorious golden era when money extracted from the surrounding islands was poured into a no expense spared building programme under Pericles who ruled the city state from 461 429 BC. The Acropolis was transformed into a magnificent city of temples with the Parthenon the temple of the city's namesake goddess Athena as its crowning glory. Literature, drama, the performing arts and philosophy all flourished during this golden age.
Jealousy from the aggressive city state of Sparta led to the Peloponnesian Wars which lasted from 431 404 BC when Sparta emerged triumphant over its arch rival Athens.
In 338 BC Athens was conquered by Philip II of Macedon and later became the power base of his son, Alexander the Great.
Athens flourished under Roman rule. Many grand buildings were constructed by various emperors notably Hadrian and the Roman nobility sent their children to study at the city's revered schools of art, literature and philosophy. But the city had relatively little importance under the Byzantine Empire which was created out of the sub division of the Roman Empire into east and west (with Byzantium present day Istanbul as the capital of the more powerful eastern region).
In 1456 Athens was captured by the Turks who held sway for the next 400 years. The subsequent resentment of the Greeks towards the Turks is still tangible today. The Turkish governor turned the Acropolis into his home and converted the Parthenon into a mosque. The city was finally freed by Greek liberators in 1827 after the six year War of Independence but the Peloponnesian town of Nafpio was initially declared the capital. Athens became the capital of Greece in 1834 under King Otho who set about transforming it into a modern city of neoclassical buildings, elegant squares and wide boulevards.
The Treaty of Lausanne in 1923 saw the forced exchange of Greeks and Turks and the arrival of nearly one million Turkish refugees in Athens. Ugly concrete apartment blocks were hastily thrown up to accommodate this sudden influx of humanity, creating the uninspiring sprawl which characterises the city outskirts today.
Athens suffered further with the German occupation in WW2 and the subsequent civil war. The military junta which was in power from 1967 74 destroyed many of the city's most beautiful old buildings including some lovely houses in the old Turkish quarter of Plaka.
The recent development of the metro, the construction of the new airport at Spata to the east of Athens and the successful hosting of the 2004 Olympics have all done much to restore the declining fortunes of this once glorious city.


text:
Q: Android - Find the orientation of photo was took by camera I need to take a photo, find out its orientation, draw it on canvas and rotate the canvas. I need help to find out orientation of the photo

A: I was searching for a similar solution to my problem.  I am taking a photo from a camera or gallery and turning it into a bitmap to use in my app.  The problem was that photos taken in PORTRAIT were being rotated -90 degrees.
Searching for answers I found this post and https://stackoverflow.com/a/11081918/3062284 which are very similar.  I actually used the latter post.  However, like @StoneHeart I was getting an error reading the Image Path.
I was using this code in my onActivityResult():
case CAMERA_REQUEST_CODE:
if (resultCode == RESULT_OK);
uriImage = data.getData();
ExifInterface exif = new ExifInterface(uriImage.getPath());

Since this error was not addressed in previous comments I thought I would share the solution I found using https://stackoverflow.com/a/10564727/3062284
I needed the real path for the file using this method from the post.
     private String getRealPathFromURI(Uri contentUri) {
        String[] proj = { MediaStore.Images.Media.DATA };
        CursorLoader loader = new CursorLoader(this, contentUri, proj, null, null, null);
        Cursor cursor = loader.loadInBackground();
        int column_index = cursor.getColumnIndexOrThrow(MediaStore.Images.Media.DATA);
        cursor.moveToFirst();
        return cursor.getString(column_index);
    }

and editing my code to the following:
case CAMERA_REQUEST_CODE:
if (resultCode == RESULT_OK);
uriImage = data.getData();
String imagePath = getRealPathFromURI(uriImage);
ExifInterface exif = new ExifInterface(imagePath);

and that fixed the "JHEAD can't open file" error I was getting.

A: You need to use the EXIF tag embedded in the photo:
private int getExifOrientation() {
  ExifInterface exif;
  int orientation = 0;
  try {
    exif = new ExifInterface( mImagePath );
    orientation = exif.getAttributeInt( ExifInterface.TAG_ORIENTATION, 1 );
  } catch ( IOException e ) {
    e.printStackTrace();
  }
  Log.d(TAG, "got orientation " + orientation);
  return orientation;
}

However, the actual EXIF value returned is sorta weird.  It allows for all variety of rotation and mirroring.  The best reference I've found is here.  In general, after you get the orientation, you'll want to run it through a lookup function to get the rotation in degrees:
private int getBitmapRotation() {
  int rotation = 0;
  switch ( getExifOrientation() ) {
    case ExifInterface.ORIENTATION_ROTATE_180:
      rotation = 180;
      break;
    case ExifInterface.ORIENTATION_ROTATE_90:
      rotation = 90;
      break;
    case ExifInterface.ORIENTATION_ROTATE_270:
      rotation = 270;
      break;
  }

  return rotation;
}

A: Can you determine orientation based on photo height and width? If it's wider than it is tall, it's landscape. If it's taller than it was wide, it's portrait.
If it's square, then you would have to read the EXIF data and hopefully it's set for those images. The EXIF data should give orientation data.

redpajama_set_name: RedPajamaStackExchange


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text:
\section{Introduction}

Quantum enveloping algebras $U_q(\mathfrak g)$ \cite{Dr86, J} of the Kac-Moody algebras $\mathfrak g$ are one of the important classes of quantum groups. Quantum enveloping algebras and their integrable highest weight representations
enjoy favorable properties, among which the canonical bases of Lusztig \cite{Lus90b} or global crystal bases of Kashiwara \cite{K91} are the most prominent ones.
Existence of such canonical bases has also been established for quantum enveloping algebras $U_q(\mathfrak g)$  of Borcherds' generalized Kac-Moody algebras $\mathfrak g$ \cite{KKK}.

On the other hand, quantum enveloping algebras have been extended to Nichols algebras of diagonal types, which include multiparameter quantum enveloping algebras as examples, in particular, one-parameter quantum enveloping algebras $U_q(\mathfrak g)$ in Kac-Moody types. In \cite{AS} Andruskiuwitch and Schneider proved that finite dimensional pointed Hopf algebras with finite abelian group (with order $>7$) of group-like elements are essentially Lusztig's small quantum groups and their variants. Furthermore, Heckenberger classified the Nicholas algebras with arithmetic root data \cite{Hec09} and proved results similar to quantum enveloping algebras (see also \cite{Hec06}).

Lusztig~\cite{Lus90a} introduced the Kostant-Lusztig ${\mathbb{A}}$-base of the quantum groups.
The ${\mathbb{A}}$-base and the finite dimensional Hopf algbebra
of the quantum group at root of unity have been key figures in study of Lusztig conjectures, (see \cite{Jan03} for history).
In this paper we establish the Kostant-Lusztig $\mathbb A$-forms for the multiparameter quantum groups and construct invariant bases for each factor of the triangular decomposition.
Our general result is based on the structure theory for multiparameter quantum groups and information on the lower rank cases, most notably
the case of $G_2$. The recent work of Fan and Li \cite{FL} on two-parameter quantum algebras has made us believe that one should be able to use our $\mathbb A$-forms to construct canonical bases for the multiparameter quantum groups.

\section{Generalized quantum groups}
In this section we recall definitions and known results about multiparameter quantum groups.
We will use the following notations throughout this paper. The ring of real numbers and integers will be denoted by ${\mathbb{R}}$ and ${\mathbb{Z}}$ respectively. For $x$, $y\in{\mathbb{R}}$, let ${\mathbb{J}}_{x,y}:=\{z\in{\mathbb{Z}}|x\leq z\leq y\}$ and ${\mathbb{J}}_{x,\infty}:=\{z\in{\mathbb{Z}}|x\leq z\}$.
Then ${\mathbb{N}}={\mathbb{J}}_{1,\infty}$ and $\bZ_{\geq 0}:={\mathbb{J}}_{0,\infty}$. Let ${\mathbb{K}}$ be a field of characteristic zero and
${\bK^\times}:={\mathbb{K}}\setminus\{0\}$. For $x$, $y\in{\mathbb{K}}$ and $r\in\bZ_{\geq 0}$, we denote $(r)_x:=\sum_{k=0}^{r-1}x^k$, $(r)_x!:=\prod_{k=1}^r(k)_x$, $(r;x,y):=1-x^{r-1}y$ and $(r;x,y)!:=\prod_{k=1}^r(k;x,y)$.

\subsection{Definition of generalized quantum groups}
Let $\theta\in{\mathbb{N}}$ and $I:={\mathbb{J}}_{1,\theta}$.
Let $V$ be a $\theta$-dimensional ${\mathbb{R}}$-linear space
with a basis $\{v_i|i\inI\}$.
Let $\defV_\bZ:=\oplus_{i\inI}{\mathbb{Z}} v_i$, so $\defV_\bZ$ is a rank-$\theta$ free ${\mathbb{Z}}$-module
(or a free abelian group).
Let $\chi:\defV_\bZ\times\defV_\bZ\to{\bK^\times}$ be a map
such that $\chi(x+y,z)=\chi(x,z)\chi(y,z)$ and $\chi(x,y+z)=\chi(x,y)\chi(x,z)$
hold for all $x$, $y$, $z\in\defV_\bZ$.
We call such $\chi$ {\it{a bi-character}}.

\begin{definition}\label{definition:defofGQG}
Let $\pi}%{\Pi:I\to\defV_\bZ$ be a map such that
$\pi}%{\Pi(I)$ is a ${\mathbb{Z}}$-base of $\defV_\bZ$.
The generalized quantum group $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ is the unique associative ${\mathbb{K}}$-algebra (with $1$) satisfying the following conditions (i)-(v).
\newline\newline
(i) As a ${\mathbb{K}}$-algebra, ${\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ is generated by
$K_\lambda$, $L_\lambda$, ($\lambda\in\defV_\bZ$), $E_i$, $F_i$
($i\inI$).
\newline\newline
(ii) The following equations hold. Let $q_{ij}:=\chi(\pi}%{\Pi(i),\pi}%{\Pi(j))$.
\begin{equation*}
\begin{array}{l}
K_0=1,\,\,K_\lambdaK_\mu=K_{\lambda+\mu}, \\
L_0=1,\,\,L_\lambdaL_\mu=L_{\lambda+\mu},  \,\,
K_\lambdaL_\mu=L_\muK_\lambda,  \\
K_{\pi}%{\Pi(i)}E_jK_{\pi}%{\Pi(i)}^{-1}=q_{ij}E_j,\,\,K_{\pi}%{\Pi(i)}F_jK_{\pi}%{\Pi(i)}^{-1}=q_{ij}^{-1}F_j, \\
L_{\pi}%{\Pi(i)}E_jL_{\pi}%{\Pi(i)}^{-1}=q_{ji}^{-1}E_j,\,\,L_{\pi}%{\Pi(i)}F_jL_{\pi}%{\Pi(i)}^{-1}=q_{ji}F_j,  \\
\mbox{$[E_i,F_j]=\delta_{ij}(-K_{\pi}%{\Pi(i)}+L_{\pi}%{\Pi(i)})$}
\end{array}
\end{equation*}
(iii) There are subspaces $U_\lambda$ ($\lambda\in\defV_\bZ$) such that
$U=\oplus_{\lambda\in\defV_\bZ}U_\lambda$,
$U_\lambdaU_\mu\subsetU_{\lambda+\mu}$,
$K_{\pi}%{\Pi(i)}^{\pm 1}\inU_0$, $L_{\pi}%{\Pi(i)}^{\pm 1}\inU_0$,
$E_i\inU_{\pi}%{\Pi(i)}$, $F_i\inU_{-\pi}%{\Pi(i)}$
\newline\newline
(iv) Let $\defU^0$ be the ${\mathbb{K}}$-subalgebra of $U$ generated by
$K_\lambdaL_\mu$ ($\lambda$, $\mu\in\defV_\bZ$).
Let $\defU^+$ (resp. $\defU^-$) be the ${\mathbb{K}}$-subalgebra of $U$ generated by
$E_i$ (resp. $F_i$)  ($i\inI$).
Then the elements $K_\lambdaL_\mu$ ($\lambda$, $\mu\in\defV_\bZ$)
form a ${\mathbb{K}}$-basis of $\defU^0$ and
we have the ${\mathbb{K}}$-linear isomorphism
${\mathbf{m}}:\defU^+\otimes\defU^0\otimes\defU^-\toU$ defined by
$X\otimes Y\otimes Z\mapsto XYZ$.
\newline\newline
(v) There exists no $X\in\defU^+\setminus\{\mathbb{K}\}$
(resp. $Y\in\defU^-\setminus\{\mathbb{K}\}$) such that
$[X,F_i]=0$ ($[E_i,Y]=0$) for all $i\inI$.
\end{definition}

Note that $\defU^0\subsetU_0$. For $\lambda\in\defV_\bZ$,
let
$\defU^+_\lambda:=\defU^+\capU_\lambda$ and $\defU^-_\lambda:=\defU^-\capU_\lambda$.
Let $\defVZ^{\defPi,+}:=\oplus_{i\in I}\bZ_{\geq 0}\pi}%{\Pi(i)$.
Then $\defU^+=\oplus_{\lambda\in\defVZ^{\defPi,+}}\defU^+_\lambda$ and
$\defU^-=\oplus_{\lambda\in\defVZ^{\defPi,+}}\defU^-_{-\lambda}$.

\begin{lemma}\label{lemma:univ}
Let ${\grave{\defU}}$ be a ${\mathbb{K}}$-algebra {\rm{(}}with $1${\rm{)}} generated by ${\grave{\defacK}}_\lambda$, ${\grave{\defacL}}_\lambda$, ${\grave{\defacE}}_i$
and ${\grave{\defacF}}_i$  satisfying conditions {\rm{(i)-(iv)}} above (in Definition~{\rm{\ref{definition:defofGQG}}}).
Then there exists a ${\mathbb{K}}$-algebra epimorphism $\xi:{\grave{U}}\toU$ such that
$\xi({\grave{\defacK}}_\lambda)=K_\lambda$, $\xi({\grave{\defacL}}_\lambda)=L_\lambda$
$(\lambda\in\defV_\bZ)$,
$\xi({\grave{\defacE}}_i)=E_i$, $\xi({\grave{\defacF}}_i)=F_i$
$(i\inI)$.
\end{lemma}
{\it{Proof.}} Let $\lambda\in\defVZ^{\defPi,+}$.
Assume that there exists $X\in{\grave{\defU}}^+_\lambda\setminus\{0\}$
such that $[X, {\grave{\defacF}}_i]=0$ for all $i\inI$.
Let ${\mathcal{I}}$ (resp. ${\mathcal{I}}^+$) be the two-sided ideal of ${\grave{\defU}}$
(resp. ${\grave{\defU}}^+$) generated by $X$. Then
${\mathcal{I}}={\rm{Span}}_{\mathbb{K}}({\grave{\defU}}^-{\grave{\defU}}^0{\mathcal{I}}^+)$.
Let ${\grave{\defU}}^\prime$ be the quotient ${\mathbb{K}}$-algebra ${\grave{\defU}}/{\mathcal{I}}$.
Then ${\grave{\defU}}^\prime$ also satisfies the same conditions as
Definition~{\rm{\ref{definition:defofGQG}}}~{\rm{(i)-(iv)}}. Let $g:{\grave{\defU}}\to {\grave{\defU}}^\prime$ be the canonical map.
Then $g_{|{\grave{\defU}}^-}:{\grave{\defU}}^-\to({\grave{\defU}}^\prime)^-$
and $g_{|{\grave{\defU}}^0}:{\grave{\defU}}^-\to({\grave{\defU}}^\prime)^0$ are the ${\mathbb{K}}$-algebra isomorphisms.
We see that $\dim ({\grave{\defU}}^\prime)^+_{\mu_1}= \dim {\grave{\defU}}^+_{\mu_1}$
for $\mu_1\in\defVZ^{\defPi,+}$ with $\mu_1-\lambda\notin\defVZ^{\defPi,+}$,
that $\dim ({\grave{\defU}}^\prime)^+_\lambda= \dim {\grave{\defU}}^+_\lambda-1$,
and that $\dim ({\grave{\defU}}^\prime)^+_{\mu_2}\leq\dim {\grave{\defU}}^+_{\mu_2}$
for $\mu_2\in\defVZ^{\defPi,+}$ with $\mu_2-\lambda\in\defVZ^{\defPi,+}$.
We also have a similar property for ${\grave{\defU}}^-$.
Then we can see the claim of this theorem by a standard argument using a direct limit. \par
\hfill $\Box$
\newline\par

By Lemma~\ref{lemma:univ}, we have the ${\mathbb{K}}$-algebra automorphism
$\defOmega^{\bhm,\defPi}:{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}\to{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ with
\begin{equation*}
\begin{array}{l}
\defOmega^{\bhm,\defPi}(K_\lambdaL_\mu)=K_{-\lambda}L_{-\mu}
\quad(\lambda,\mu\in\defV_\bZ), \\
\defOmega^{\bhm,\defPi}(E_i)=F_iL_{-\pi}%{\Pi(i)},\,\,
\defOmega^{\bhm,\defPi}(F_i)=K_{-\pi}%{\Pi(i)}E_i
\quad(i\inI).
\end{array}
\end{equation*}

Define the bicharacter $\bhm^{\mathrm{op}}:\defV_\bZ\times\defV_\bZ\to{\bK^\times}$ by
$\bhm^{\mathrm{op}}(x,y):=\chi(y,x)$
($x$, $y\in\defV_\bZ$).
By Lemma~\ref{lemma:univ},
we have the ${\mathbb{K}}$-algebra isomorphism
$\defUpsilon^{\bhmop,\defPi}: U(\bhm^{\mathrm{op}},\pi}%{\Pi)\to{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ with
\begin{align}\label{e:chevalley}
\defUpsilon^{\bhmop,\defPi}(K_\lambdaL_\mu)&=K_\muL_\lambda,\,(\lambda,\mu\in\defV_\bZ),\\
\defUpsilon^{\bhmop,\defPi}(E_i)&=F_i,\,
\defUpsilon^{\bhmop,\defPi}(F_i)=E_i
\,(i\inI),
\end{align}
which will be referred as the {\it Chevalley involution}. We also have the ${\mathbb{K}}$-algebra anti-automorphism
$\defGamma^{\bhm^{\mathrm{op}},\defPi}:{\defU(\bhm^{\mathrm{op}},\defPi)}\to{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ with
\begin{equation*}
\defGamma^{\bhm^{\mathrm{op}},\defPi}(K_\lambdaL_\mu)=K_\muL_\lambda,\,(\lambda,\mu\in\defV_\bZ),\,\,
\defGamma^{\bhm^{\mathrm{op}},\defPi}(E_i)=E_i,\,\defGamma^{\bhm^{\mathrm{op}},\defPi}(F_i)=F_i\,(i\inI).
\end{equation*}

For $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$, we have
the following equations. See \cite{Hec10} for the
notation ${{m}\choose{r}}_{q_{ii}}$ and ${{k}\choose{r}}_{q_{ii}}$.
\begin{equation*}
\begin{array}{lcl}
E_i^kF_i^m &
=& \sum_{r=0}^{{\mathrm{min}}\{k,m \}}(r)_{q_{ii}}!{m\choose r}_{q_{ii}}
{k\choose r}_{q_{ii}}
q_{ii}^{{\frac {r(-2k+r+1)} 2}} \\
& & \quad\quad\quad\quad\quad\cdot(\prod_{s=0}^{r-1}(-K_{\pi}%{\Pi(i)}+q_{ii}^{-m+k+s}L_{\pi}%{\Pi(i)}))
F_i^{m-r}E_i^{k-r},
\end{array}
\end{equation*} whence
\begin{equation*}
\begin{array}{l}
\mbox{$[E_i,F_i^m]$}=(m)_{q_{ii}}(-K_{\pi}%{\Pi(i)}+q_{ii}^{-m+1}L_{\pi}%{\Pi(i)})F_i^{m-1}, \\
\mbox{$[F_i,E_i^m]$}=(m)_{q_{ii}}(-L_{\pi}%{\Pi(i)}+q_{ii}^{-m+1}K_{\pi}%{\Pi(i)})E_i^{m-1}.
\end{array}
\end{equation*}
For $m\in\bZ_{\geq 0}$ and $i$, $j\inI$ with $i\ne j$,
define $E_{m,i,j}$, $\defacE^\vee_{m,i,j}\in\defU^+_{\pi}%{\Pi(j)+m\pi}%{\Pi(i)}$, $F_{m,i,j}$
$F_{m,i,j}\in\defU^-_{-\pi}%{\Pi(j)-m\pi}%{\Pi(i)}$
inductively by
$E_{0,i,j}:=\defacE^\vee_{0,i,j}:=E_j$,
$F_{0,i,j}:=\defacE^\vee_{0,i,j}:=F_j$, and
\begin{equation}\label{eqn:EFp}
\begin{array}{l}
E_{m+1,i,j}:=E_iE_{m,i,j}
-q_{ii}^mq_{ij}E_{m,i,j}E_i,\,
\defacE^\vee_{m+1,i,j}:=\defacE^\vee_{m,i,j}E_i
-q_{ii}^mq_{ji}E_i\defacE^\vee_{m,i,j},
\\
F_{m+1,i,j}:=F_iF_{m,i,j}
-q_{ii}^mq_{ji}F_{m,i,j}F_i,\,
\defacF^\vee_{m+1,i,j}:=\defacF^\vee_{m,i,j}F_i
-q_{ii}^mq_{ij}F_i\defacF^\vee_{m,i,j}.
\end{array}
\end{equation}
We have
\begin{equation}\label{eqn:defUpEmij}
\begin{array}{l}
\defUpsilon^{\bhmop,\defPi}(E_{m,i,j})=F_{m,i,j}, \quad
\defUpsilon^{\bhmop,\defPi}(F_{m,i,j})=E_{m,i,j}, \\
\defUpsilon^{\bhmop,\defPi}(\defacE^\vee_{m,i,j})=\defacF^\vee_{m,i,j}, \quad
\defUpsilon^{\bhmop,\defPi}(\defacF^\vee_{m,i,j})=\defacE^\vee_{m,i,j}.
\end{array}
\end{equation}
We have
\begin{equation}\label{eqn:tUbiddtwo}
\begin{array}{l}
\mbox{$[E_i,F_{m,i,j}]$}=-(m)_{q_{ii}}
(m;q_{ii},q_{ij}q_{ji})K_{\pi}%{\Pi(i)}F_{m-1,i,j},
\\
\mbox{$[F_i,E_{m,i,j}]$}=-(m)_{q_{ii}}
(m;q_{ii},q_{ij}q_{ji})L_{\pi}%{\Pi(i)}E_{m-1,i,j},
\\
\mbox{$[E_j, F_{m,i,j}]$}=
(m;q_{ii},q_{ij}q_{ji})!F_i^mL_{\pi}%{\Pi(j)},
\\
\mbox{$[F_j, E_{m,i,j}]$}=
(m;q_{ii},q_{ij}q_{ji})!E_i^mK_{\pi}%{\Pi(j)},
\\
\mbox{$[E_{m,i,j},F_{m,i,j}]$}=(m)_{q_{ii}}!
(m;q_{ii},q_{ij}q_{ji})!(-K_{\pi}%{\Pi(j)+m\pi}%{\Pi(i)}+L_{\pi}%{\Pi(j)+m\pi}%{\Pi(i)}),
\\
\mbox{$[E_{m,i,j},F_{m^\prime,i,k}]$}=0
\quad (m^\prime\in\bZ_{\geq 0},k\inI\setminus\{i,j\}).
\end{array}
\end{equation} We have
\begin{equation}\label{eqn:OPiEFrij}
\begin{array}{l}
\defOmega^{\bhm,\defPi}(E_{r,i,j})=q_{ii}^{-{\frac {r(r-1)} {2}}}q_{ji}^{-r}
F_{r,i,j}L_{-\pi}%{\Pi(j)-r\pi}%{\Pi(i)}, \\
\defOmega^{\bhm,\defPi}(\defacE^\vee_{r,i,j})=q_{ii}^{-{\frac {r(r-1)} {2}}}q_{ij}^{-r}
\defacF^\vee_{r,i,j}L_{-\pi}%{\Pi(j)-r\pi}%{\Pi(i)}, \\
\defOmega^{\bhm,\defPi}(F_{r,i,j})=q_{ii}^{\frac {r(r-1)} {2}}q_{ji}^r
K_{-\pi}%{\Pi(j)-r\pi}%{\Pi(i)}E_{r,i,j}, \\
\defOmega^{\bhm,\defPi}(\defacF^\vee_{r,i,j})=q_{ii}^{\frac {r(r-1)} {2}}q_{ij}^r
K_{-\pi}%{\Pi(j)-r\pi}%{\Pi(i)}\defacE^\vee_{r,i,j}.
\end{array}
\end{equation} We have
\begin{equation}\label{eqn:XiPiEFrij}
\begin{array}{l}
\defGamma^{\bhm^{\mathrm{op}},\defPi}\defGamma^{\bhm,\defPi}={\mathrm{id}}_{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)},\,\defGamma^{\bhm,\defPi}\defGamma^{\bhm^{\mathrm{op}},\defPi}={\mathrm{id}}_{\defU(\bhm^{\mathrm{op}},\defPi)},  \\
\defGamma^{\bhm^{\mathrm{op}},\defPi}(E_{r,i,j})=\defacE^\vee_{r,i,j},\,
\defGamma^{\bhm^{\mathrm{op}},\defPi}(F_{r,i,j})=\defacF^\vee_{r,i,j}.
\end{array}
\end{equation}

\subsection{Kharchenko's Poincar{\'{e}}-Birkohoff-Witt theorem and Heckenberger's Lusztig~isomorphisms}
We recall the following theorem by Kharchenko. We also introduce some notations.

\begin{theorem}\label{theorem:KharchenkoPBW}
{\rm{(\cite{Kha99})}}
Let $\chi$ be a bicharacter and $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ be the generalized quantum group.
Then there exists a unique pair $(\defRbhm^{\defPi,+},\varphi_\bhm^{\defPi,+})$ of a
subset $\defRbhm^{\defPi,+}$ of $\defVZ^{\defPi,+}$ and a map $\varphi_\bhm^{\defPi,+}:\defRbhm^{\defPi,+}\to{\mathbb{N}}$
satisfying the following:

Let $X:=\{(\alpha,t)\in\defRbhm^{\defPi,+}\times{\mathbb{N}}|t\in{\mathbb{J}}_{1,\varphi_\bhm^{\defPi,+}(\alpha)}\}$.
Define the map $z:X\to\defRbhm^{\defPi,+}$ by $z(\alpha,t):=\alpha$.
Let $Y$ be the set of maps $y:X\to\bZ_{\geq 0}$ such that
$|\{x\in X|y(x)\geq 1\}|<\infty$ and
$(y(x))_{\chi(z(x),z(x))}!\ne 0$ for all
$x\in X$. Then
\newline\newline
${\rm{(*)}}$\quad $\forall\lambda\in\defVZ^{\defPi,+}$,
$\dim\defU^+_\lambda=|\{y\in Y|\sum_{x\in X}y(x)z(x)=\lambda\}|$.
\newline\newline Moreover, letting $q_{i^\prime j^\prime}:=\chi(\pi}%{\Pi(i^\prime),\pi}%{\Pi(j^\prime))$
$(i^\prime, j^\prime\inI)$, for $i$, $j\inI$ with $i\ne j$, we have
\begin{equation}\label{eqn:goodbhmcond}
\{t\in\bZ_{\geq 0}|\pi}%{\Pi(j)+t\pi}%{\Pi(i)\in\defRbhm^{\defPi,+}\}=\{t\in\bZ_{\geq 0}|(t)_{q_{ii}}!(t;q_{ii},q_{ij}q_{ji})!\ne 0\}.
\end{equation}
\end{theorem}

Let $\defRbhm^\defPi:=\defRbhm^{\defPi,+}\cup(-\defRbhm^{\defPi,+})$.
Define the map $\varphi_\bhm^\defPi:\defRbhm^\defPi\to{\mathbb{N}}$
by $\varphi_\bhm^\defPi(-\alpha):=\varphi_\bhm^\defPi(\alpha):=\varphi_\bhm^{\defPi,+}(\alpha)$
($\alpha\in\defRbhm^{\defPi,+}$).
We say that $\chi$ is {\it{$(\pi}%{\Pi,i)$-good}} if it satisfies the condition that
for every $j\inI\setminus\{i\}$,
there exists $N^{\bhm,\defPi}_{ij}\in\bZ_{\geq 0}$ such that
$\pi}%{\Pi(j)+N^{\bhm,\defPi}_{ij}\pi}%{\Pi(i)\in\defRbhm^{\defPi,+}$ and
$\pi}%{\Pi(j)+(N^{\bhm,\defPi}_{ij}+1)\pi}%{\Pi(i)\notin\defRbhm^{\defPi,+}$.
Let $N^{\bhm,\defPi}_{ii}:=-2$.

Let $i\inI$ and $\chi$ be a $(\pi}%{\Pi,i)$-good bicharacter.
Define the map ${\deftaubhm_i\defPi}:I\to\defV_\bZ$
by ${\deftaubhm_i\defPi}(j):=\pi}%{\Pi(j)+N^{\bhm,\defPi}_{ij}\pi}%{\Pi(i)$
($j\inI$). By \eqref{eqn:goodbhmcond},
we see that ${\deftaubhm_i\defPi}$ is a $(\pi}%{\Pi,i)$-good bicharacter, and
\begin{equation}\label{eqn:Nbhminv}
N^{\bhm,\tauidefPi}_{ij}=N^{\bhm,\defPi}_{ij}\quad(j\inI),
\end{equation} whence $(\tau^\bhm_i)^2\pi}%{\Pi=\pi}%{\Pi$.

\begin{theorem}\label{theorem:HecIso}
{\rm{(\cite{Hec10})}}
Let $a:I\to{\bK^\times}$ be a map.
Let $i\inI$. Let $\chi$ be a $(\pi}%{\Pi,i)$-good bicharacter.
Then there exists a
${\mathbb{K}}$-algebra isomorphism
$T^{\chi,{\deftaubhm_i\defPi},a}_i:{\defU(\bhm,\tauidefPi)}\to{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ such that
\begin{equation}\label{eqn:dfLusT}
\begin{array}{l}
T_i(K_\lambda)=K_\lambda,\quad T_i(L_\lambda)=L_\lambda
\quad (\lambda\in\defV_\bZ),
\\
T_i(E_i)=a(i)F_iL_{-\pi}%{\Pi(i)},\quad
T_i(F_i)={\frac 1 {a(i)}}K_{-\pi}%{\Pi(i)}E_i,\\
T_i(E_j)=a(j)E_{N^{\bhm,\defPi}_{ij},i,j} \quad (j\inI\setminus\{i\})\\
T_i(F_j)={\frac 1
{a(j)(N^{\bhm,\defPi}_{ij})_{q_{ii}}!(N^{\bhm,\defPi}_{ij};q_{ii},q_{ij}q_{ji})!}}
F_{N^{\bhm,\defPi}_{ij},i,j} \quad (j\inI\setminus\{i\}),
\end{array}
\end{equation}
where $T_i:=T^{\chi,{\deftaubhm_i\defPi},a}_i$ and $q_{i^\prime j^\prime}:=\chi(\pi}%{\Pi(i^\prime),\pi}%{\Pi(j^\prime))$
$(i^\prime,\,j^\prime\inI)$. Moreover we have
\begin{equation}\label{eqn:HecIsoEq}
\defRbhm^\tauidefPi=\defRbhm^\defPi,\,\,\defRbhm^{\tauidefPi,+}=(\defRbhm^{\defPi,+}\setminus\{\pi}%{\Pi(i)\})\cup\{-\pi}%{\Pi(i)\},\,\,
\varphi_\bhm^\defPi=\varphi_\bhm^\tauidefPi.
\end{equation}
\end{theorem}

Let $i$, $j\inI$ be such that $i\ne j$.
Let $\chi$ be a $(\pi}%{\Pi,i)$-good and $(\pi}%{\Pi,j)$-good bicharacters.
We say that $\chi$ is {\it{a $(\pi}%{\Pi,i,j)$-good bicharacter}}
if $\tau^\bhm_{i_1}\cdots\tau^\bhm_{i_m}\pi}%{\Pi$
can be defined for all $m\in{\mathbb{N}}$ and
all $i_t\in\{i,j\}$
($t\in{\mathbb{J}}_{1,m}$).

\begin{equation}\label{eqn:defmij}
\mbox{Let $m^{\bhm,\defPi}_{ij}:=|\defRbhm^{\defPi,+}\cap({\mathbb{Z}}\pi}%{\Pi(i)\oplus{\mathbb{Z}}\pi}%{\Pi(j))|(\in{\mathbb{J}}_{2,\infty}\cup\{\infty\})$.}
\end{equation}

By \eqref{eqn:HecIsoEq}, using the same argument as that for \cite[Lemma~4]{HY08}, we have the following result.
\begin{lemma}\label{lemma:HecIRktetwo}
Let $\chi$ be a $(\pi}%{\Pi,i,j)$-good bicharacter.
Let $X:={\mathbb{Z}}\pi}%{\Pi(i)\oplus{\mathbb{Z}}\pi}%{\Pi(j)$.
Let $m:=m^{\bhm,\defPi}_{ij}$.
Let $i_{2y-1}:=i$, $i_{2y}:=j$ $(y\in{\mathbb{N}})$.
Let $\pi}%{\Pi_t:=\tau^\bhm_{i_t}\tau^\bhm_{i_{t-1}}\cdots\tau^\bhm_{i_1}\pi}%{\Pi$
$(t\in{\mathbb{N}})$. Then for
$k\in{\mathbb{J}}_{1,m}$, we have
\begin{equation*}
|\defRbhm^{\defPi,+}\cap(-R_\chi^{\pi}%{\Pi_k,+})|=|\defRbhm^{\defPi,+}\cap(-R_\chi^{\pi}%{\Pi_k,+})\cap X|=k.
\end{equation*}
Moreover if $m<\infty$, we have
\begin{equation*}
\pi}%{\Pi_m=\tau^\bhm_{i_{m+1}}\tau^\bhm_{i_m}\cdots\tau^\bhm_{i_2}\pi}%{\Pi,
\end{equation*} and
\begin{equation*}
\defRbhm^{\defPi,+}\cap X=\{\pi}%{\Pi(i_1)=i,\pi}%{\Pi_1(i_2),\ldots,\pi}%{\Pi_{m-2}(i_{m-1}),
\pi}%{\Pi_{m-1}(i_m)=j\}.
\end{equation*}
Furthermore $|\{\pi}%{\Pi_t(i_{t+1})|t\in{\mathbb{N}}\}|=\infty$ if $m=\infty$.
\end{lemma}

Let $\chi$ be a $(\pi}%{\Pi,i,j)$-good bicharacter.
We say that $\chi$ is {\it{a $(\pi}%{\Pi,i,j)$-good finite bicharacter}}
if $m$ in Lemma~\ref{lemma:HecIRktetwo} is finite.

\begin{theorem}\label{theorem:HecIfund} {\rm{(\cite{Hec10})}}
Let $i, j \in I, i\neq j$ and $\chi$ be a $(\pi}%{\Pi,i,j)$-good finite bicharacter with $m:=m^{\bhm,\defPi}_{ij} <\infty$.
For $t\in{\mathbb{N}}$, let $\pi}%{\Pi^\prime_t:=\tau^\bhm_{i_{t+1}}\tau^\bhm_{i_t}\cdots\tau^\bhm_{i_2}\pi}%{\Pi$
{\rm{(}}$\mbox{$i_k = i$ {\rm{(}}{\rm resp.} $j${\rm{)}}}$ if $k$ is odd {\rm{(}}resp. even{\rm{)}}{\rm{)}},
 $a_t$, $a^\prime_t:I\to{\bK^\times}$ be maps,
${\dot{T}}_{(t)}:=T^{\chi,\pi}%{\Pi_1,a_1}_{i_1}\cdotsT^{\chi,\pi}%{\Pi_t,a_t}_{i_t}$
and ${\ddot{T}}_{(t)}:=T^{\chi,\pi}%{\Pi^\prime_1,a^\prime_1}_{i_2}\cdotsT^{\chi,\pi}%{\Pi^\prime_t,a^\prime_t}_{i_{t+1}}$.
Then there exists a map $b:I\to{\bK^\times}$ such that
\begin{equation*}
{\dot{T}}_{(m)}(E_k)=b(k){\ddot{T}}_{(m)}(E_k),\,\,
{\dot{T}}_{(m)}(F_k)={\frac 1 {b(k)}}{\ddot{T}}_{(m)}(F_k)
\quad (k\inI).
\end{equation*}
Moreover there exists $z\in{\bK^\times}$ such that
\begin{equation*}
{\dot{T}}_{(m-1)}(E_{i_m})=zE_j,\,\,{\dot{T}}_{(m-1)}(F_{i_m})=
z^{-1}F_j.
\end{equation*}
\end{theorem}

\subsection{Strict Heckenberger's Lusztig isomorphisms}

\begin{lemma}\label{lemma:plusminusone}
Let $i, j \in I, i\neq j$ and $\chi$ be a $(\pi}%{\Pi,i,j)$-good finite bicharacter with $m:=m^{\bhm,\defPi}_{ij} <\infty$. Also
$b:I\to{\bK^\times}$ be the map and $z\in{\bK^\times}$  in Theorem~{\rm{\ref{theorem:HecIfund}}}. Then the following statements hold.
 \newline\newline
{\rm{(1)}} If $\defOmega^{\bhm,\defPi}{\dot{T}}_{(m)}={\dot{T}}_{(m)}\Omega^{\chi,\pi}%{\Pi_m}$,
then $b(k)\in\{1,-1\}$ for all $k\inI$.\newline
{\rm{(2)}} If $\defOmega^{\bhm,\defPi}{\dot{T}}_{(m-1)}={\dot{T}}_{(m-1)}\Omega^{\chi,\pi}%{\Pi_{m-1}}$,
then $z\in\{1,-1\}$.
\end{lemma}
{\it{Proof.}} (1) Let $k\inI$. We have
\begin{equation}
\begin{array}{l}
F_k=b(k){\ddot{T}}_{(m)}^{-1}{\dot{T}}_{(m)}(F_k)
=b(k){\ddot{T}}_{(m)}^{-1}{\dot{T}}_{(m)}\Omega^{\chi,\pi}%{\Pi_m}(E_k
L_{\pi}%{\Pi(k)}) \\
\quad =b(k)\defOmega^{\bhm,\defPi}{\ddot{T}}_{(m)}^{-1}{\dot{T}}_{(m)}(E_k
L_{\pi}%{\Pi(k)})
=b(k)^2\defOmega^{\bhm,\defPi}(E_kL_{\pi}%{\Pi(k)})
=b(k)^2F_k,
\end{array}
\end{equation} whence $b(k)^2=1$, so $b(k)\in\{1,-1\}$. \newline\par
(2) This can be proved similarly to (1).
\hfill $\Box$.

\begin{lemma}\label{lemma:OmegaT}
Let $a:I\to{\bK^\times}$ be a map.
Let $i\inI$ and $\chi$ be a $(\pi}%{\Pi,i)$-good bicharacter.
Then
\begin{equation*}
\defOmega^{\bhm,\defPi}T^{\chi,{\deftaubhm_i\defPi},a}_i
=T^{\chi,{\deftaubhm_i\defPi},a}_i\Omega^{\chi,{\deftaubhm_i\defPi}}
\end{equation*} if and only if
\begin{equation}\label{eqn:dfOmegaT}
a(i)^2=1\quad\mbox{and}\quad a(j)^2={\frac {q_{ii}^{\frac {N^{\bhm,\defPi}_{ij}(N^{\bhm,\defPi}_{ij}-1)} 2}q_{ji}^{N^{\bhm,\defPi}_{ij}}}
{(N^{\bhm,\defPi}_{ij})_{q_{ii}}!(N^{\bhm,\defPi}_{ij};q_{ii},q_{ij}q_{ji})!}}\,\,
(j\inI\setminus\{i\}).
\end{equation}
\end{lemma}
{\it{Proof.}} The claim follows from Theorem~\ref{theorem:HecIso} and \eqref{eqn:OPiEFrij}
\hfill $\Box$

\begin{lemma}\label{lemma:sltwoac}
Let $G$ be the ${\mathbb{K}}$-algebra {\rm{(}}with $1${\rm{)}} defined with
the generators $X$, $Y$, $Z$ and the relations
$[Z,X]=2X$, $[Z,Y]=-2Y$, $[X,Y]=Z$.
{\rm{(}}Namely  $G$ is isomorphic to the universal enveloping algebra
of ${\mathrm{sl}}_2({\mathbb{K}})$.{\rm{)}}
Let $k\in\bZ_{\geq 0}$. Let $\Gamma$ be the
$(k+1)$-dimensional left $G$-module
with the ${\mathbb{K}}$-basis $\{\gamma_r|r\in{\mathbb{J}}_{0,k}\}$
such that $Z\gamma_r:=(k-2x)\gamma_r$,
$Y\gamma_r:=\gamma_{r+1}$, $X\gamma_r:=x(k-r+1)\gamma_{r-1}$,
where $\gamma_{-1}:=\gamma_{k+1}:=0$.
Let $a\in{\bK^\times}$. Define $\xi\in{\rm{End}}_{\mathbb{K}}(\Gamma)$
by $\xi(v):=\exp(aX)\exp(-a^{-1}Y)\exp(aX)\cdot v$
$(v\in\Gamma)$. Then
\begin{equation*}\label{eqn:gYrmo}
\xi(Y^r\gamma_0)={\frac{(-1)^{k-r}a^{2r-k}r!} {(k-r)!}}Y^{k-r}\gamma_0, \quad
\xi(X^r\gamma_k)={\frac{(-1)^ra^{k-2r}r!} {(k-r)!}}X^{k-r}\gamma_k.
\end{equation*}
\end{lemma}
{\it{Proof.}} Let ${\bar \Gamma}$, ${\bar \gamma}_y$ ($y\in{\mathbb{J}}_{0,1}$),
${\bar \xi}$ be $\Gamma$, $\gamma_y$, $\xi$ respectively for $k=1$.
Then ${\bar \xi}({\bar \gamma}_0)=(-a^{-1}){\bar \gamma}_1$ and ${\bar \xi}({\bar \gamma}_1)=a{\bar \gamma}_0$.
Regard ${\bar \Gamma}^{\otimes k}$ as the ($k$-fold tensor) $G$-module
in a standard way. Let $g:M\to{\bar M}^{\otimes k}$ be the $G$-module monomorphism $g:M\to{\bar \Gamma}^{\otimes k}$
such that $g(\gamma_0)={\bar \gamma}_0^{\otimes k}$.
Then
\begin{equation*}
g(\gamma_x)=g(Y^x\gamma_0)=Y^x\cdot g(\gamma_0)=r!\sum_{|\{x\in{\mathbb{J}}_{1,k}|i_x=1\}|=x}{\bar \gamma}_{i_1}\otimes\cdots\otimes{\bar \gamma}_{i_1}.
\end{equation*}
Note $g(\xi(v))={\bar \xi}^{\otimes k}(g(v))$ ($v\in\Gamma$).
Then
\begin{equation*}
\begin{array}{l}
g(\xi(\gamma_r))=g(t(Y^r\gamma_0))={\bar \xi}^{\otimes k}(g(Y^r\gamma_0))\\
\quad ={\frac{(-a^{-1})^{k-r}a^rr!} {(k-r)!}}g(Y^{k-r}\gamma_0)
=g({\frac{(-1)^{k-r}a^{2r-k}r!} {(k-r)!}}\gamma_{k-r}).
\end{array}
\end{equation*}
Thus we can see the claim.
\hfill $\Box$

\begin{lemma}\label{lemma:ranktwot}
Let $a_{11}:=a_{22}:=2(\in{\mathbb{K}})$. Let
$a_{12}$, $a_{21}\in{\mathbb{K}}$ be such that
$(a_{12},a_{21})\in\{(0,0),(-1,-1),(-2,-1),(-3,-1)\}$.
Let $m\in\bZ_{\geq 0}$ be $2$ {\rm{(}}{\rm{resp.}} $3$, {\rm{resp.}} $4$, {\rm{resp.}} $6${\rm{)}}
if $a_{12}$ is $0$ {\rm{(}}{\rm{resp.}} $-1$, {\rm{resp.}} $-2$, {\rm{resp.}} $-3${\rm{)}}.
Let $i_{2x-1}:=1$, $i_{2x}:=2$ $(x\in{\mathbb{N}})$.
Let $G$ be a ${\mathbb{K}}$-algebra {\rm{(}}with $1${\rm{)}}
satisfying the following conditions {\rm{(i)}} and {\rm{(ii)}}.
\newline\newline
{\rm{(i)}}
There exist $X_i$, $Y_i$, $Z_i\in G$
$(i\in{\mathbb{J}}_{1,2})$ such that
\begin{equation*}
\begin{array}{l}
[Z_1,Z_2]=0,\,[Z_i,X_j]=a_{ij}X_j,\,[Z_i,Y_j]=-a_{ij}Y_j,\,[X_i,Y_j]=\delta_{ij}Z_i, \\
({\mathrm{ad}} X_{i^\prime})^{1-a_{i^\prime j^\prime}}(X_{j^\prime})
=({\mathrm{ad}} Y_{i^\prime})^{1-a_{i^\prime j^\prime}}(Y_{j^\prime})=0\,\,(i^\prime\ne j^\prime).
\end{array}
\end{equation*}
\newline
{\rm{(ii)}} For $\Re\in G$ and $\Im\in\{X_i,Y_i|i\in{\mathbb{J}}_{1,2}\}$, there exists $r\in{\mathbb{N}}$
such that $({\mathrm{ad}} \Im)^r(\Re)=0$.
\newline\newline
For $i\in{\mathbb{J}}_{1,2}$, let $b_i\in{\bK^\times}$ $(i\in{\mathbb{J}}_{1,2})$, and
define the ${\mathbb{K}}$-algebra automorphism $\xi_i$ by
$\xi_i(\Re):=\exp(b_i{\mathrm{ad}} X_i)\exp(-b_i^{-1}{\mathrm{ad}} Y_i)\exp(b_i{\mathrm{ad}} X_i)(\Re)$
$(\Re\in G)$.
Then we have
\begin{equation}\label{eqn:rktwota}
\begin{array}{l}
\xi_i(X_i)=-b_i^2Y_i,\,\,\xi_i(Y_i)=-b_i^{-2}X_i, \\
\xi_i(X_j)={\frac {b_i^{-a_{ij}}} {(-a_{ij})!}}({\mathrm{ad}} X_i)^{-a_{ij}}(X_j),\,\,
\xi_i(Y_j)={\frac {(-b_i)^{a_{ij}}} {(-a_{ij})!}}({\mathrm{ad}} Y_i)^{-a_{ij}}(Y_j) \quad (i\ne j),
\end{array}
\end{equation}
\begin{equation}\label{eqn:rktwotb}
\begin{array}{l}
\xi_{i_t}\xi_{i_{t+1}}\cdots\xi_{i_{t+m-2}}(X_{i_{t+m-1}})=X_{i_{t-1}}, \\
\xi_{i_t}\xi_{i_{t+1}}\cdots\xi_{i_{t+m-2}}(Y_{i_{t+m-1}})=Y_{i_{t-1}} \\
(t\in{\mathbb{Z}}),
\end{array}
\end{equation} and
\begin{equation}\label{eqn:rktwotc}
\xi_{i_1}\xi_{i_2}\cdots\xi_{i_m}
=\xi_{i_2}\xi_{i_3}\cdots\xi_{i_{m+1}}.
\end{equation}
\end{lemma}
{\it{Proof.}} We can see \eqref{eqn:rktwota} by Lemma~\ref{lemma:sltwoac}.
As for \eqref{eqn:rktwotb},
for example, if $a_{12}=-3$, by Lemma~\ref{lemma:sltwoac}, we have
\begin{equation*}
\begin{array}{l}
\xi_1\xi_2\xi_1\xi_2\xi_1(X_2) \\
\quad = \xi_1\xi_2\xi_1\xi_2({\frac {b_1^3} {3!}}[X_1,[X_1,[X_1,X_2]]]) \\
\quad = -{\frac {b_1^3} {3!}}\xi_1\xi_2\xi_1\xi_2([X_1,[X_1,[X_2,X_1]]]) \\
\quad = -{\frac {b_1^3} {3!}}\xi_1\xi_2\xi_1(
[b_2[X_2,X_1],[b_2[X_2,X_1],(-1)b_2^{-1}X_1]]) \\
\quad = -{\frac {b_1^3b_2} {3!}}\xi_1\xi_2(
[(-1){\frac {b_1} {2!}}[X_1,[X_1,X_2]],{\frac {2!} {b_1}}[X_1,X_2]]) \\
\quad ={\frac {b_1^3b_2} {3!}}\xi_1\xi_2(
[[X_1,[X_2,X_1]],[X_2,X_1]]) \\
\quad ={\frac {b_1^3b_2} {3!}}\xi_1(
[[b_2[X_2,X_1],(-1)b_2^{-1}X_1],(-1)b_2^{-1}X_1]) \\
\quad =-{\frac {b_1^3} {3!}}\xi_1([X_1,[X_1,[X_1,X_2]]]) \\
\quad =(-1){\frac {b_1^3} {3!}}\cdot (-1){\frac {3!} {b_1^3}}X_2 \\
\quad =X_2.
\end{array}
\end{equation*}
Now we show \eqref{eqn:rktwotc}. Let $\xi^\prime:=\xi_{i_1}\xi_{i_2}\cdots\xi_{i_{m-1}}$.
By \eqref{eqn:rktwotb}, for $\Re\in G$, we see
\begin{equation*}
\begin{array}{l}
\xi^\prime\xi_{i_m}(\xi^\prime)^{-1}(\Re) \\
\quad =\exp(b_i{\mathrm{ad}} \xi^\prime(X_{i_m}))\exp(-b_i^{-1}{\mathrm{ad}} \xi^\prime(Y_{i_m}))
\exp(b_i{\mathrm{ad}} \xi^\prime(X_{i_m}))(\Re) \\
\quad =\xi_{i_2}(\Re).
\end{array}
\end{equation*}
This completes the proof. \hfill $\Box$
\newline\par
\begin{definition}\label{definition:admbhm}
Let $\chi:\defV_\bZ\times\defV_\bZ\to{\bK^\times}$ be a bi-character, $\pi}%{\Pi:I\to\defV_\bZ$ be a map such that
$\pi}%{\Pi(I)$ is a ${\mathbb{Z}}$-base of $\defV_\bZ$ and $q_{ij}:=\chi(\pi}%{\Pi(i),\pi}%{\Pi(j))$ for all $i,j \in I$.
Let $A=[a_{ij}]_{ij\inI}$ be a symmetrizable generalized Cartan matrix.
Let $d_i\in{\mathbb{N}}$ be such that $d_ia_{ij}=d_ja_{ji}$.
($i$, $j\inI$).
Let ${\dot{q}}\in{\bK^\times}$ be such that
${\dot{q}}^r\ne 1$ for all $r\in{\mathbb{N}}$.
Let $\sqrt{\bhm}:\defV_\bZ\times\defV_\bZ\to{\bK^\times}$ be a bicharacter and ${\dot{q}}_{ij}:=\sqrt{\bhm}(\pi}%{\Pi(i),\pi}%{\Pi(j))$ ($i$, $j\inI$).
Assume that ${\dot{q}}_{ii}={\dot{q}}^{2d_i}$ $(i\inI)$,
${\dot{q}}_{ij}{\dot{q}}_{ji}={\dot{q}}^{2d_ia_{ij}}$ $(i,j\inI)$,
$q_{ij}:={\dot{q}}_{ij}^2$ for all $i$, $j\inI$.
Also assume that for every $k\inI$, there exists $\Theta(q_{kk}-1)\in{\bK^\times}$ such that
$\Theta(q_{kk}-1)^2=q_{kk}-1$.
Then we say that such $\chi$ is {\it{a $(\pi}%{\Pi,A)$-admissible bicharacter}}.
If $A$ is the Cartan matrix of a finite-dimensional complex Lie algebra
(i.e., $A$ is a symmetrizable generalized Cartan matrix of finite-type),
we call ${\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ {\it{a finite-type multiparameter quantum group.}}
\end{definition}

For a $(\pi}%{\Pi,A)$-admissible bicharacter $\chi$, ${\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ is presented by the generators given by
Definition~\ref{definition:defofGQG}~(i)
and the relations composed of those of Definition~\ref{definition:defofGQG}~(ii)
and $E_{1-a_{ij},i,j}=F_{1-a_{ij},i,j}=0$ ($i,j\inI$, $i\ne j$),
which is well-known and can be proved by a standard argument along with Theorem~\ref{theorem:LusPBW} below.

\begin{lemma} \label{lemma:lmfundaTh}
Let $\chi$ be a $(\pi}%{\Pi,A=[a_{ij}]_{ij\inI})$-admissible bicharacter.
\newline\newline {\rm{(1)}}  Then $\chi$ is $(\pi}%{\Pi,i)$-good bi-character for every $i\inI$
and $N^{\bhm,\defPi}_{ij}=-a_{ij}$ for all $i$, $j\inI$.
\newline {\rm{(2)}}
For $i\inI$ let
${\ddot{q}}_{jk}:=\sqrt{\bhm}({\deftaubhm_i\defPi}(j),{\deftaubhm_i\defPi}(k))$
$(j,k\inI)$.
Then we have
\begin{equation*}
{\ddot{q}}_{jj}={\dot{q}}_{jj}
\,\,(j\inI),\,\,
{\ddot{q}}_{jk}{\ddot{q}}_{kj}
={\dot{q}}_{jk}{\dot{q}}_{kj}\,\,(j,k\inI).
\end{equation*}
In particular, $\chi$ is $({\deftaubhm_i\defPi},A)$-admissible.
\newline {\rm{(3)}} For $i$, $j\inI$ with $i\ne j$,
$\chi$ is $(\pi}%{\Pi,i,j)$-finite if and only if
$a_{ij}a_{ji}\in{\mathbb{J}}_{0,3}$. Moreover
\begin{equation}\label{eqn:prmij}
m^{\bhm,\defPi}_{ij}=
\left\{\begin{array}{ll}
2 & \mbox{if $a_{ij}a_{ji}=0$,} \\
3 & \mbox{if $a_{ij}a_{ji}=1$,} \\
4 & \mbox{if $a_{ij}a_{ji}=2$,} \\
6 & \mbox{if $a_{ij}a_{ji}=3$.}
\end{array}\right.
\end{equation}
\end{lemma}
{\it{Proof.}} Claim~(1) follows from \eqref{eqn:goodbhmcond}.
Claim~(2) can be proved directly.
Claim~(3) follows from Claims~(1) and (2) and Lemma~\ref{lemma:HecIRktetwo}. \hfill $\Box$
\newline\par

Let $\chi$ be a $(\pi}%{\Pi,A=[a_{ij}]_{ij\inI})$-admissible bicharacter.
Let
${\dot{q}}_{jk}$ be as in Definition~\ref{definition:admbhm}. Define the map
$\varpi:I\to{\bK^\times}$ by
\begin{equation*}
\varpi(j)=
\left\{\begin{array}{ll}
1 & \quad (j=i), \\
{\frac {{\dot{q}}_{ij}^{a_{ij}}}
{(-a_{ij})_{q_{ii}}!\Theta(q_{ii}-1)^{-a_{ij}}}} & \quad(j\inI\setminus\{i\}).
\end{array}\right.
\end{equation*}
Let
\begin{equation*}
\LusT^{\bhm,\tauidefPi}_i:=T^{\chi,{\deftaubhm_i\defPi},\varpi}_i.
\end{equation*}
We see directly that $\varpi$ satisfies \eqref{eqn:dfOmegaT}.
As for \eqref{eqn:dfLusT}, letting $T_i:=\LusT^{\bhm,\tauidefPi}_i$, we have
\begin{equation*}
\begin{array}{l}
T_i(K_\lambdaL_\mu)=K_\lambdaL_\mu \,
(\lambda,\mu\in\defV_\bZ),\,\,
T_i(E_i)=F_iL_{-\pi}%{\Pi(i)},\,T_i(F_i)=K_{-\pi}%{\Pi(i)}E_i, \\
T_i(\defacE^\vee_{r,i,j})={\frac {(r)_{q_{ii}}!{\dot{q}}_{ij}^{a_{ij}+2r}\Theta(q_{ii}-1)^{a_{ij}+2r}}
{(-a_{ij}-r)_{q_{ii}}!}}E_{-a_{ij}-r,i,j},\\
T_i(\defacF^\vee_{r,i,j})={\frac {(r)_{q_{ii}}!{\dot{q}}_{ii}^{(a_{ij}+2r)(a_{ij}-1)}{\dot{q}}_{ij}^{-a_{ij}-2r}\Theta(q_{ii}-1)^{a_{ij}+2r}}
{(-a_{ij}-r)_{q_{ii}}!}}F_{-a_{ij}-r,i,j}
\\ (j\inI\setminus\{i\},\,r\in{\mathbb{J}}_{0,-a_{ij}}).
\end{array}
\end{equation*} We also have
\begin{equation} \label{eqn:InvLusTi}
(\LusT^{\bhm,\defPi}_i)^{-1}=\defGamma^{\bhm^{\mathrm{op}},\defPi}\LusT^{\bhmop,\tauidefPi}_i\defGamma^{\bhm,{\deftaubhm_i\defPi}}.
\end{equation}

For $i\inI$, define the ${\mathbb{K}}$-algebra automorphism
$\zeta^{\bhm,\defPi}_i:{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}\to{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ by
\begin{equation*}
\begin{array}{l}
\zeta^{\bhm,\defPi}_i(K_\lambdaL_\mu):=K_\lambdaL_\mu\,\,
(\lambda, \mu\in\defV_\bZ), \\
\zeta^{\bhm,\defPi}_i(E_j):={\frac 1 {{\dot{q}}_{ij}{\dot{q}}_{ji}}}E_j,\,
\zeta^{\bhm,\defPi}_i(F_j):={\dot{q}}_{ij}{\dot{q}}_{ji}F_j\,\,
(j\inI).
\end{array}
\end{equation*} Then we have
\begin{equation*}\label{eqn:Lustzata}
\LusT^{\bhm,\tauidefPi}_i\Upsilon^{\bhm^{\mathrm{op}},{\deftaubhm_i\defPi}}=\zeta^{\bhm,\defPi}_i\defUpsilon^{\bhmop,\defPi}
\LusT^{\bhmop,\tauidefPi}_i\quad(i\inI).
\end{equation*}
For $i\inI$, let
\begin{equation*}
{\bar{\defacE}}_i:={\frac {E_i} {\Theta(q_{ii}-1)}},\quad
{\bar{\defacF}}_i:=-{\frac {F_i} {\Theta(q_{ii}-1)}},\quad
{\bar{H}}_i:={\frac {K_{\pi}%{\Pi(i)}-L_{\pi}%{\Pi(i)}} {q_{ii}-1}}.
\end{equation*}
Note that
\begin{equation*}
[{\bar{\defacE}}_i,{\bar{\defacF}}_i]=\delta_{ij}{\bar{H}}_i,
\end{equation*} and
\begin{equation*}
{\bar{H}}_i{\bar{\defacE}}_j=q_{ij}{\bar{\defacE}}_j{\bar{H}}_i+q_{ij}^{-1}
{\frac {q_{ij}q_{ji}-1} {q_{ii}-1}}{\bar{\defacE}}_jL_{\pi}%{\Pi(i)}.
\end{equation*}

Let ${\mathbb{O}}$ be
the ${\mathbb{Q}}$-subalgebra of ${\mathbb{K}}$ generated by
${\dot{q}}^{\pm 1}_{ij}$, ${\frac 1 {(-a_{ij})_{q_{ii}}!}}$ for all $i,j\inI$.
Let $\defU_\bO$ (resp. $\defUo_\bO$, resp. $\defUp_\bO$, resp. $\defUm_\bO$)
be the ${\mathbb{O}}$-subalgebra (with $1$) of $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ (resp. $\defU^0$, resp. $\defU^+$, resp. $\defU^-$)
generated by
$K_{\pi}%{\Pi(i)}^{\pm 1}$,  $L_{\pi}%{\Pi(i)}^{\pm 1}$, ${\bar{H}}_i$, ${\bar{\defacE}}_i$, ${\bar{\defacF}}_i$
(resp. $K_{\pi}%{\Pi(i)}^{\pm 1}$,  $L_{\pi}%{\Pi(i)}^{\pm 1}$, ${\bar{H}}_i$,
resp. ${\bar{\defacE}}_i$, resp. ${\bar{\defacF}}_i$) for all $i\inI$.
We can see
$\defU_\bO=\defUm_\bO\otimes_{\mathbb{O}}\defUo_\bO\otimes_{\mathbb{O}}\defUp_\bO$.
We also see that the elements
\begin{equation*}
\prod_{i\inI}K_{\pi}%{\Pi(i)}^{x(i)}(K_{\pi}%{\Pi(i)}L_{\pi}%{\Pi(i)})^{y(i)}
{\bar{H}}_i^{z(i)}
\end{equation*}
($x(i)\in\{0,1\}$, $z(i)\in\bZ_{\geq 0}$, $y(i)\in{\mathbb{Z}}$) form
${\mathbb{O}}$-basis of $\defUo_\bO$.

\begin{theorem} \label{theorem:fundaTh}
Let $i, j \in I, i\neq j$ and $\chi$ be a $(\pi}%{\Pi,i,j)$-good finite bicharacter with $m:=m^{\bhm,\defPi}_{ij} <\infty$.
Assume that $\chi$ is $(\pi}%{\Pi,A)$-admissible. Let $b:I\to{\bK^\times}$ and $z\in{\bK^\times}$ be as in Theorem~{\rm{\ref{theorem:HecIfund}}}.
Assume that $T^{\chi,\pi}%{\Pi_t,a_t}_{i_t}=T^{\chi,\pi}%{\Pi_t}_{i_t}$
and $T^{\chi,\pi}%{\Pi^\prime_t,a^\prime_t}_{i_{t+1}}=T^{\chi,\pi}%{\Pi^\prime_t}_{i_{t+1}}$
$(t\in{\mathbb{N}})$. Then we have
\begin{equation}\label{eqn:maineq}
b(k)=1\,\,(k\inI)\quad{and}\quad{z=1}.
\end{equation}
\end{theorem}
{\it{Proof.}}
We divide the proof into steps.

{\it{Step}}~1. {\it{Assume that ${\dot{q}}$ is transcendental over ${\mathbb{Q}}$
and that ${\dot{q}}_{j^\prime i^\prime}=1$
for all $i^\prime$, $j^\prime\inI$ with $i^\prime < j^\prime$.}}
Then ${\mathbb{O}}$ is the
${\mathbb{Q}}$-subalgebra of ${\mathbb{K}}$ generated by
${\dot{q}}^{\pm 1}$, ${\frac 1 {(-a_{i^\prime j^\prime})_{q_{i^\prime i^\prime}}!}}$
for all $i^\prime$, $j^\prime\inI$ with $i^\prime \ne j^\prime$.
So ${\mathbb{O}}$ is a principal integral domain.

Consider the ${\mathbb{Q}}$-algebra $(\defU_\bO)_1:=\defU_\bO/({\dot{q}}-1)\defU_\bO$.
Let $f:\defU_\bO\to (\defU_\bO)_1$ be the canonical map.
Note that for $k\inI$, we have $f(L_{\pi}%{\Pi(k)})=f(K_{\pi}%{\Pi(k)})$,
and we see that $f(K_{\pi}%{\Pi(k)})$ is a central element of $(\defU_\bO)_1$.
So we can consider the quotient ${\mathbb{Q}}$-algebra $(\defU_\bO)_1/(f(K_{\pi}%{\Pi(k)})-1)$.
By Lemmas~\ref{lemma:plusminusone}, \ref{lemma:OmegaT} and \ref{lemma:ranktwot},
we have \eqref{eqn:maineq}.

{\it{Step}}~2. Let $U^\prime$ denote the $U$ of Step~1, and
let ${\ddot{q}}$ denote ${\dot{q}}$ for $U^\prime$.
Then ${\ddot{q}}$ is transcendental over ${\mathbb{Q}}$.
Assume that that ${\dot{q}}_{j^\prime i^\prime}=1$
for all $i^\prime$, $j^\prime\inI$ with $i^\prime < j^\prime$.
We use a specialization argument with ${\ddot{q}}\to{\dot{q}}$;
this ${\dot{q}}$ is the one for $U$ of this step.
We can obtain \eqref{eqn:maineq}
from Step~1 by considering
the ${\mathbb{O}}$-subalgebra of $U^\prime$
generating by $K_{\pi}%{\Pi(t)}^{\pm 1}$,  $L_{\pi}%{\Pi(t)}^{\pm 1}$,
$E_t$, $F_t$ ($t\inI$)
and using Lemma~\ref{lemma:univ}.

{\it{Step}}~3. {\it{General cases.}}
Repeat the same arguments as in Step~2.
\hfill $\Box$

\subsection{Drinfeld pairing}

Let $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ be the generalized quantum group.
We regard $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ as a Hopf algebra $(U,\HopfD,\HopfS,\Hopfe)$
with
$\HopfD(K_\lambda)=K_\lambda\otimesK_\lambda$,
$\HopfD(L_\lambda)=L_\lambda\otimesL_\lambda$,
$\HopfD(E_i)=E_i\otimes  1+K_{\pi}%{\Pi(i)}\otimesE_i$,
$\HopfD(F_i)=F_i\otimes
L_{\pi}%{\Pi(i)}+1\otimesF_i$,
$\HopfS(K_\lambda)=K_{-\lambda}$,
$\HopfS(L_\lambda)=L_{-\lambda}$,
$\HopfS(E_i)=-K_{-\pi}%{\Pi(i)}E_i$,
$\HopfS(F_i)=-F_iL_{-\pi}%{\Pi(i)}$,
$\Hopfe(K_\lambda)=\Hopfe(L_\lambda)=1$,
and $\Hopfe(E_i)=\Hopfe(E_i)=0$.

For $i$, $j\inI$ with $i\ne j$ and $r\in\bZ_{\geq 0}$, if $E_{
r,i,j}\ne 0$, we have
\begin{equation*}
\HopfD(E_{r,i,j})=E_{r,i,j}\otimes 1
+\sum_{k=0}^r
{\frac {(r)_{q_{ii}}!(k;q_{ii},q_{ii}^{r-k}q_{ij}q_{ji})!} {(k)_{q_{ii}}!(r-k)_{q_{ii}}!}}
E_i^kK_{(r-k)\pi}%{\Pi(i)+\pi}%{\Pi(j)}\otimesE_{r-k,i,j}.
\end{equation*}

Let $\defU^{+,\flat}=\tBp(\bhm,\defPi):=\oplus_{\lambda\in\defV_\bZ}\defU^+K_\lambda$,
and $\defU^{-,\flat}=\tBm(\bhm,\defPi):=\oplus_{\lambda\in\defV_\bZ}\defU^-L_\lambda$.
Then $U={\mathrm{Span}}_{\mathbb{K}}(\defU^{-,\flat}\defU^{+,\flat})={\mathrm{Span}}_{\mathbb{K}}(\defU^{+,\flat}\defU^{-,\flat})$ and in a
standard way (see \cite{Dr86}), we have a bilinear form
$\vartheta=\vartheta^{\bhm,\defPi}:\defU^{+,\flat}\times\defU^{-,\flat}\to{\mathbb{K}}$ with the following
properties:
\begin{equation}\label{eqn:bprtoftB}
\begin{array}{l}
\vartheta(K_\lambda,L_\mu)=\chi(\lambda,\mu),
\vartheta(E_i,F_j)=\delta_{ij},
\vartheta(K_\lambda,F_j)=\vartheta(E_i,L_\lambda)=0, \\
\vartheta(X^+Y^+,X^-)=
\sum_{k^-}\vartheta(X^+,(X^-)^{(2)}_{k^-})\vartheta(Y^+,(X^-)^{(1)}_{k^-}),\\
\vartheta(X^+,X^-Y^-)=
\sum_{k^+}\vartheta((X^+)^{(1)}_{k^+},X^-)\vartheta((X^+)^{(2)}_{k^+},Y^-),\\
\vartheta(\HopfS(X^+),X^-)=\vartheta(X^+,\HopfS^{-1}(X^-)),\\
\vartheta(X^+,1)=\Hopfe(X^+),
\vartheta(1,X^-)=\Hopfe(X^-), \\
X^-X^+  =\sum_{r^+,r^-}
\vartheta((X^+)^{\prime,(1)}_{r^+}, \HopfS((X^-)^{\prime,(1)}_{r^-}))
\vartheta((X^+)^{\prime,(3)}_{r^+}, (X^-)^{\prime,(3)}_{r^-}) \\
\quad\quad\quad\quad\quad\quad\quad\quad
\cdot(X^+)^{\prime,(2)}_{r^+}(X^-)^{\prime,(2)}_{r^-}, \\
X^+X^- =\sum_{r^+,r^-}
\vartheta((X^+)^{\prime,(3)}_{r^+}, \HopfS((X^-)^{\prime,(3)}_{r^-}))
\vartheta((X^+)^{\prime,(1)}_{r^+}, (X^-)^{\prime,(1)}_{r^-}) \\
\quad\quad\quad\quad\quad\quad\quad\quad
\cdot(X^-)^{\prime,(2)}_{r^-}(X^+)^{\prime,(2)}_{r^+}
\end{array}
\end{equation} for $\lambda$, $\mu\in\defV_\bZ$, $i$, $j\inI$,
and $X^+$, $Y^+\in\defU^{+,\flat}$, $X^-$, $Y^-\in\defU^{-,\flat}$,
where $(X^+)^{(x)}_{k^+}$ and $(X^-)^{(x)}_{k^-}$ with $x\in{\mathbb{J}}_{1,2}$
(resp. $(X^+)^{\prime,(y)}_{r^+}$ and
$(X^-)^{\prime,(y)}_{r^-}$ with $y\in{\mathbb{J}}_{1,3}$)
are any elements of $\defU^{+,\flat}$ and $\defU^{-,\flat}$ respectively
satisfying $\HopfD(X^\pm)=\sum_{k^\pm}(X^\pm)^{(1)}_{k^\pm}
\otimes (X^\pm)^{(2)}_{k^\pm}$,
(resp. $(({\mathrm{id}}_U\otimes\HopfD)\circ\HopfD)(X^\pm)=
\sum_{r^\pm}(X^\pm)^{\prime,(1)}_{r^\pm}\otimes
(X^\pm)^{\prime,(2)}_{r^\pm}\otimes (X^\pm)^{\prime,(3)}_{r^\pm}$).

We have
\begin{equation*}
\begin{array}{l}
\vartheta^{\bhm,\defPi}(X^+K_\lambda,X^-L_\mu)=\vartheta^{\bhm,\defPi}(X^+,X^-)\chi(\lambda,\mu)
\\
\quad (X^+\in\defU^+,\,X^-\in\defU^-,\,\lambda,\mu\in\defV_\bZ).
\end{array}
\end{equation*}
It follows that $\vartheta^{\bhm,\defPi}_{|\defU^+\times\defU^-}$ is non-degenerate.
We have $\vartheta^{\bhm,\defPi}(X^+,X^-)=0$ for $\lambda,\mu\in\defVZ^{\defPi,+}$ with
$\lambda\ne\mu$ and $X^+\in\defU^+_\lambda$,
$X^-\in\defU^-_{-\mu}$.

\section{Kostant-Lusztig ${\mathbb{A}}$-form}  \label{section:LAform}

In this section we establish the Kostant-Lusztig $\mathbb A$-forms for the
finite-type multiparameter
quantum group $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$ (see Definitions~\ref{definition:defofGQG} and \ref{definition:admbhm}) and construct invariant bases for each factor of the triangular decomposition. Let $A=[a_{ij}]_{ij\inI}$ be a Cartan matrix associated with a finite dimensional complex simple Lie algebra,
i.e., $A$ is a symmetrizable generalized Cartan matrix of finite-type. We assume that $\chi$ is a $(\pi}%{\Pi,A)$-admissible bicharacter
(see Definition~\ref{definition:admbhm}). Recall the symbols $q_{ij}$, ${\dot{q}}_{ij}$
and $\Theta(q_{ii}-1)$. Let $W$ be the Weyl group associated to the Cartan matrix $A$ and generated by the simple reflections
$s_i$ ($i\inI$) with relations
$s_i^2=e$ ($i\inI$) and $(s_is_j)^{m^{\bhm,\defPi}_{ij}}=e$,
($e$ is the identity element of $W$). Note that $W$ is the finite Weyl group.

\subsection{Some standard notations and results} \label{subsection:somno}
Define the map $\ell:W\to\bZ_{\geq 0}$ by $\ell(e):=0$
and $\ell(w):=\min\{r\in{\mathbb{N}}|\exists i_t\inI\,(t\in{\mathbb{J}}_{1,r}), w=s_{i_1}\cdotss_{i_r}\}$.
In fact $\ell$ is the length map of the Coxeter system $(W,\{s_i|i\inI\})$.
It is well-known that
\begin{equation*}
\mbox{$|\ell(ws_i)-\ell(w)|=1$ for $w\inW$ and $i\inI$.}
\end{equation*} Let $W$ act on $\defV_\bZ$ by
$s_i\cdot\pi}%{\Pi(j):=\pi}%{\Pi(j)-a_{ij}\pi}%{\Pi(j)$ ($i,j\inI$). Use the convention as follows
\begin{equation*}
\mbox{Let $s_{i_1}\cdotss_{i_t}$ (resp. $\tau^\bhm_{i_{t-1}}\cdots\tau^\bhm_{i_1}\pi}%{\Pi$)
mean $e$ (resp. $\pi}%{\Pi$) if $t=0$.}
\end{equation*}
We have
\begin{equation*}
s_{i_1}\cdotss_{i_{t-1}}\cdot\pi}%{\Pi(i_t)=\tau^\bhm_{i_{t-1}}\cdots\tau^\bhm_{i_1}\pi}%{\Pi(i_t)
\,\,(t\in{\mathbb{N}},\,i_x\inI\,(x\in{\mathbb{J}}_{1,t})).
\end{equation*}

For $w=s_{k_1}\ldotss_{k_r}\inW$, if $r=\ell(w)$,
the expression $s_{k_1}\ldotss_{k_{\ell(w)}}$ is called {\it{reduced}}.
For $w=s_{k_1}\ldotss_{k_{\ell(w)}}\inW$,
let
\begin{equation*}
\rmid^{\bhm,\defPi}\LusT_w:=T_{k_1}^{\chi,\pi}%{\Pi_1}\cdotsT_{k_{\ell(w)}}^{\chi,\pi}%{\Pi_{\ell(w)}}\quad(\rmid^{\bhm,\defPi}\LusT_e:={\mathrm{id}}_{{\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}}),
\end{equation*}
where $\pi}%{\Pi_t:=\tau^\bhm_{k_t}\cdots\tau^\bhm_{k_1}\pi}%{\Pi$
($t\in{\mathbb{J}}_{1,{\ell(w)}}$). By Theorem~\ref{theorem:fundaTh}, $\rmid^{\bhm,\defPi}\LusT_w$ is independent of the
choice of reduced expressions for $w$.

 It is well-known that there exists a unique
$w_\circ\inW$ such that $\ell(w)\leq\defell(\twoh)$ for all $w\inW$;
$w_\circ$ is called {\it{the longest element}}.
We also know that $\defell(\twoh)=|\defRbhm^{\defPi,+}|$ and
\begin{equation}\label{eqn:prtwoh}
\forall w\inW,\,\,\defell(\twoh)=\ell(w)+\ell(w^{-1}w_\circ).
\end{equation}

Let $n=(n_1,\ldots,n_{\defell(\twoh)})\inI^{\defell(\twoh)}$ be such that $s_{n_1}\cdotss_{n_{\defell(\twoh)}}=w_\circ$
(reduced expression of $w_\circ$).
For $t\in{\mathbb{J}}_{1,\defell(\twoh)}$, let $\beta_{n;t}:=s_{n_1}\cdotss_{n_{t-1}}\cdot\pi}%{\Pi(n_t)$.
By
\eqref{eqn:HecIsoEq} and Lemma~\ref{lemma:lmfundaTh}, we have (see \cite[1.7]{Hum90})
\begin{equation*}
\defRbhm^{\defPi,+}=\{\beta_{n;t}|t\in{\mathbb{J}}_{1,\defell(\twoh)}\}.
\end{equation*}
Thus $\defRbhm^\defPi$ can be identified with the root system of $W$.
For $t\in{\mathbb{J}}_{1,\defell(\twoh)}$, let
\begin{equation*}
\begin{array}{l}
E_{n;t}:=\rmid^{\bhm,\defPi}\LusT_{s_{n_1}\ldots s_{n_{t-1}}}(E_{n_t}),\,\,
F_{n;t}:=\rmid^{\bhm,\defPi}\LusT_{s_{n_1}\ldots s_{n_{t-1}}}(F_{n_t}), \\
{\bar{\defacE}}_{n;t}:=\rmid^{\bhm,\defPi}\LusT_{s_{n_1}\ldots s_{n_{t-1}}}({\bar{\defacE}}_{n_t}),\,\,
{\bar{\defacF}}_{n;t}:=\rmid^{\bhm,\defPi}\LusT_{s_{n_1}\ldots s_{n_{t-1}}}({\bar{\defacF}}_{n_t}),
\end{array}
\end{equation*} ($E_{n;1}:=E_{n_1}$,
$F_{n;1}:=F_{n_1}$, ${\bar{\defacE}}_{n;1}:={\bar{\defacE}}_{n_1}$,
${\bar{\defacF}}_{n;1}:={\bar{\defacF}}_{n_1}$), that is,
\begin{equation*}
{\bar{\defacE}}_{n;t}={\frac {E_{n;t}} {\Theta(q_{n_tn_t}-1)}}
={\frac {E_{n;t}} {\Theta(\chi(\beta_{n;t},\beta_{n;t})-1)}}.
\end{equation*}
Let $n_0\inI$ be such that $s_{n_0}w_\circ s_{n_{\defell(\twoh)}}=w_\circ$.
By Theorem~\ref{theorem:fundaTh}, using a standard argument (see \cite[Proposition~8.20]{Jan96}), we have
\begin{equation} \label{eqn:Enlwo}
E_{n;\defell(\twoh)}=E_{n_0},\,
F_{n;\defell(\twoh)}=F_{n_0},\,
{\bar{\defacE}}_{n;\defell(\twoh)}={\bar{\defacE}}_{n_0},\,
{\bar{\defacF}}_{n;\defell(\twoh)}={\bar{\defacF}}_{n_0}.
\end{equation}

The following result can be proved by  a standard argument (see \cite{AY2015}, \cite{HY10} for example).

\begin{theorem}  \label{theorem:LusPBW}
Let $k:=\defell(\twoh)$, $J:={\mathbb{J}}_{1,k}$
and $\beta_t:=s_{n_1}\cdotss_{n_{t-1}}\cdot\pi}%{\Pi(n_t)$
$(t\in J)$.
\newline\newline
{\rm{(1)}} Let $\sigma:J\to J$
be a bijection. Then the elements
\begin{equation*}
E_{n;\sigma(1)}^{x_1}\cdotsE_{n;\sigma(k)}^{x_k}\quad(x_t\in\bZ_{\geq 0}\,(t\in J)))
\end{equation*} form a ${\mathbb{K}}$-basis of $\defU^+$.
\newline
{\rm{(2)}} Let $y$, $z\in J$ be such that $y<z$.
Let $X$ be a ${\mathbb{K}}$-subalgebra of $U$
generated by the elements $E_{n;x}$ {\rm{(}}$x\in{\mathbb{J}}_{y+1,z-1}${\rm{)}}. Then
\begin{equation}\label{eqn:EnyEnz}
E_{n;y}E_{n;z}-\chi(\beta_y,\beta_z)E_{n;z}E_{n;y}\in X.
\end{equation}
\newline
{\rm{(3)}} We have
\begin{equation*}
\vartheta^{\bhm,\defPi}(E_{n;k}^{x_k}\cdotsE_{n;1}^{x_1},F_{n;k}^{y_k}\cdotsF_{n;1}^{y_1})
=\prod_{t=1}^k\delta_{x_t,y_t}(x_t)_{\chi(\beta_t,\beta_t)}!,
\end{equation*} where $x_t$, $y_t\in\bZ_{\geq 0}$ $(t\in J)$.
\end{theorem}

\subsection{Type ${\mathrm{G}}_2$} \label{subsection:typeGtwo}
In this subsection we assume $\theta=2$ (so $I={\mathbb{J}}_{1,2}$), and $A=[a_{ij}]_{ij\inI}$ is the Cartan
matrix of type $G_2$. So
 $a_{12}=-3$ and $a_{21}=-1$.
Consider the generalized quantum group $U={\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$.
Let $q_{ij}$ be as in Definition~\ref{definition:defofGQG} and $\chi$ be a $(\pi}%{\Pi,A)$-admissible bicharacter.
Let $q:=q_{11}$ and $a:=q_{12}$.
Then $q_{22}=q^3$ and $q_{21}=q^{-3}a^{-1}$.
Since $E_{4,1,2}=E_{2,2,1}=0$, we have
\begin{equation*}
\begin{array}{l}
E_1^4E_2-(1+q)(1+q^2)aE_1^3E_2E_1+q(1+q^2)(1+q+q^2)a^2E_1^2E_2E_1^2 \\
\quad -q^3(1+q)(1+q^2)a^3E_1^2E_2E_1^2+q^6a^4E_2E_1^4=0, \\
E_1^2E_2-(1+q)(1-q+q^2)aE_2E_1E_2+q^3a^2E_2^2E_1=0.
\end{array}
\end{equation*}

Let
\begin{equation*}
\begin{array}{l}
K_1;=K_{\pi}%{\Pi(1)},\,\,K_2;=K_{\pi}%{\Pi(2)}, \,\,
 E_{12}:=E_{1,1,2},\,\,
 E_{112}:=E_{2,1,2},\\
 E_{1112}:=E_{3,1,2},\,\,
 E_{11212}:=E_{112}E_{12}-aq^2E_{12}E_{112}.
\end{array}
\end{equation*}
Then we have
\begin{equation*}
\begin{array}{l}
E_{12}E_2=aq^3E_2E_{12},\,\,
E_{1112}E_2=a^3q^6E_2E_{1112}+a^2q^3(q^2-1)(q-1)E_{12}^3, \\
E_{112}E_2=a^2q^3E_2E_{112}+aq(q^2-1)E_{12}^2, \\
E_{1112}E_2=a^3q^3E_2E_{1112}+aq(q^2-q-1)E_{11212}
+a^2q^2(q^3-1)E_{12}E_{112}, \\
E_1E_2=aE_2E_1+E_{12},\,\,
E_{11212}E_{12}=aq^3E_{12}E_{11212}, \\
E_{112}E_{12}=aq^2E_{12}E_{112}+E_{11212},\,\,
E_{1112}E_{12}=a^2q^3E_{12}E_{1112}+
{\frac {aq(q^3-1)} {q+1}}E_{112}^2, \\
E_1E_{12}=aqE_{12}E_1+E_{112},\,\,
E_{112}E_{11212}=aq^3E_{112}E_{11212}, \\
E_{1112}E_{11212}=a^3q^6E_{11212}E_{1112}+
{\frac {a^2q^3(q^3-1)(q-1)} {q+1}}E_{112}^3, \\
E_1E_{11212}=a^2q^3E_{11212}E_1+
{\frac {aq(q^3-1)} {q+1}}E_{112}^2,\,\,
E_{1112}E_{112}=aq^3E_{112}E_{1112}, \\
E_1E_{112}=aq^2E_{112}E_1+E_{1112},\,\,
E_1E_{1112}=aq^3E_{1112}E_1.
\end{array}
\end{equation*}
We have
\begin{equation*}
\begin{array}{l}
\HopfD(E_{12})=E_{12}\otimes 1+(1-q^{-3})E_1K_2\otimesE_2
+K_1K_2\otimesE_{12}, \\
\HopfD(E_{112})=E_{112}\otimes 1+(1-q^{-3})(1-q^{-2})E_1^3K_2\otimesE_2 \\
\quad\quad\quad\quad\quad +(1-q^{-2})(1+q)E_1K_1K_2\otimesE_{12}+K_1^2K_2\otimesE_{112}, \\
\HopfD(E_{1112})=E_{1112}\otimes 1+(1-q^{-3})(1-q^{-2})(1-q^{-1})E_1^3K_2\otimesE_2 \\
\quad\quad\quad\quad\quad +(q^2-1)(1-q^{-3})E_1^2K_1K_2\otimesE_{12}, \\
\quad\quad\quad\quad\quad +q^{-1}(q^3-1)E_1K_1^2K_2\otimesE_{112}+K_1^3K_2\otimesE_{1112},
\end{array}
\end{equation*}
and
\begin{equation*}
\begin{array}{l}
\HopfD(E_{11212})=E_{11212}\otimes 1+{\frac {(q^3-1)^2} {q^4}}E_{112}E_1K_2\otimesE_2
+{\frac {(q^3-1)(q^2-q-1)} {aq^5}}E_{1112}K_2\otimesE_2 \\
\quad\quad\quad\quad\quad\quad +{\frac {q^3-1} {q}}E_{112}K_1K_2\otimesE_{12}
+{\frac {(q^3-1)^2(q^2-1)(q-1)} {aq^{12}}}E_1^3K_1^2\otimesE_2^2 \\
\quad\quad\quad\quad\quad\quad +{\frac {(q^3-1)^2(q^2-1)} {q^6}}E_1^2K_1K_2\otimesE_2E_{12}
+{\frac {(q^3-1)(q^2-1)} {q^3}}E_1K_1^2K_2^2\otimesE_{12}^2 \\
\quad\quad\quad\quad\quad\quad +K_1^3K_2^2\otimesE_{11212}.
\end{array}
\end{equation*}
Let
\begin{equation*}
\begin{array}{l}
{\hat{\defacE}}_1:=E_1,\,{\hat{\defacE}}_2:=E_2,\,{\hat{\defacE}}_{12}:={\frac {q^3} {q^3-1}}E_{12},
\,{\hat{\defacE}}_{112}:={\frac {q^5} {(q^3-1)(q^2-1)}}E_{112}, \\
{\hat{\defacE}}_{1112}:={\frac {q^6} {(q^3-1)(q^2-1)(q-1)}}E_{1112},\,
{\hat{\defacE}}_{11212}:={\frac {q^9} {(q^3-1)^2(q^2-1)(q-1)}}E_{11212}.
\end{array}
\end{equation*} Then we have
\begin{equation*}
\begin{array}{l}
{\hat{\defacE}}_{1112}{\hat{\defacE}}_2=a^3q^6{\hat{\defacE}}_2{\hat{\defacE}}_{1112}+a^2q^3(q^3-1){\hat{\defacE}}_{12}^3,\,
{\hat{\defacE}}_{112}{\hat{\defacE}}_2=a^2q^3{\hat{\defacE}}_2{\hat{\defacE}}_{112}+a(q^3-1){\hat{\defacE}}_{12}^2, \\
{\hat{\defacE}}_{1112}{\hat{\defacE}}_2=a^3q^3{\hat{\defacE}}_2{\hat{\defacE}}_{1112}+aq^{-2}(q^2-q-1)(q^3-1){\hat{\defacE}}_{11212} \\
\quad\quad\quad\quad\quad +a^2(q^3-1)(q^2+q+1){\hat{\defacE}}_{12}{\hat{\defacE}}_{112}, \\
{\hat{\defacE}}_1{\hat{\defacE}}_2=a{\hat{\defacE}}_2{\hat{\defacE}}_1+q^{-3}(q^3-1){\hat{\defacE}}_{12},\,
{\hat{\defacE}}_{112}{\hat{\defacE}}_{12}=aq^2{\hat{\defacE}}_{12}{\hat{\defacE}}_{112}+q^{-1}(q-1){\hat{\defacE}}_{11212}, \\
{\hat{\defacE}}_{1112}E_{12}=a^2q^3{\hat{\defacE}}_{12}{\hat{\defacE}}_{1112}+a(q^3-1){\hat{\defacE}}_{112}^2,\,
{\hat{\defacE}}_1E_{12}=aq{\hat{\defacE}}_{12}{\hat{\defacE}}_1+q^{-2}(q^2-1){\hat{\defacE}}_{112}, \\
{\hat{\defacE}}_{1112}{\hat{\defacE}}_{11212}=a^3q^6{\hat{\defacE}}_{11212}{\hat{\defacE}}_{1112}+
a^2q^3(q^3-1){\hat{\defacE}}_{112}^3, \\
{\hat{\defacE}}_1{\hat{\defacE}}_{11212}=a^2q^3{\hat{\defacE}}_{11212}{\hat{\defacE}}_1+
a(q^3-1){\hat{\defacE}}_{112}^2,\,
{\hat{\defacE}}_1{\hat{\defacE}}_{112}=aq^2{\hat{\defacE}}_{112}{\hat{\defacE}}_1+q^{-1}(q-1){\hat{\defacE}}_{1112}.
\end{array}
\end{equation*} and
\begin{equation*}
\begin{array}{l}
\HopfD({\hat{\defacE}}_{12})={\hat{\defacE}}_{12}\otimes 1+{\hat{\defacE}}_1K_2\otimes{\hat{\defacE}}_2
+K_1K_2\otimes{\hat{\defacE}}_{12}, \\
\HopfD({\hat{\defacE}}_{112})={\hat{\defacE}}_{112}\otimes 1+{\hat{\defacE}}_1^3K_2\otimes{\hat{\defacE}}_2
+(q+1){\hat{\defacE}}_1K_1K_2\otimes{\hat{\defacE}}_{12}+K_1^2K_2\otimesE_{112}, \\
\HopfD({\hat{\defacE}}_{1112})={\hat{\defacE}}_{1112}\otimes 1+{\hat{\defacE}}_1^3K_2\otimes{\hat{\defacE}}_2
+(q^2+q+1){\hat{\defacE}}_1^2K_1K_2\otimes{\hat{\defacE}}_{12}, \\
\quad\quad\quad\quad\quad +(q^2+q+1){\hat{\defacE}}_1K_1^2K_2\otimes{\hat{\defacE}}_{112}+K_1^3K_2\otimes{\hat{\defacE}}_{1112}, \\
\HopfD({\hat{\defacE}}_{11212})={\hat{\defacE}}_{11212}\otimes 1
+(q^2+q+1){\hat{\defacE}}_{112}{\hat{\defacE}}_1K_2\otimes{\hat{\defacE}}_2 \\
\quad\quad\quad\quad\quad\quad +a^{-1}q^{-2}(q^2-q-1){\hat{\defacE}}_{1112}K_2\otimes{\hat{\defacE}}_2 \\
\quad\quad\quad\quad\quad\quad +(q^2+q+1){\hat{\defacE}}_{112}K_1K_2\otimes{\hat{\defacE}}_{12}
+a^{-1}q^{-3}{\hat{\defacE}}_1^3K_1^2\otimes{\hat{\defacE}}_2^2 \\
\quad\quad\quad\quad\quad\quad +(q^2+q+1){\hat{\defacE}}_1^2K_1K_2\otimes{\hat{\defacE}}_2{\hat{\defacE}}_{12}
+(q^2+q+1){\hat{\defacE}}_1K_1^2K_2^2\otimes{\hat{\defacE}}_{12}^2 \\
\quad\quad\quad\quad\quad\quad +K_1^3K_2^2\otimes{\hat{\defacE}}_{11212}.
\end{array}
\end{equation*}

For ${\bf a}=(a_1,\ldots,a_6)\in\bZ_{\geq 0}^6$, let
\begin{equation*}
\begin{array}{l}
Q_1({\bf a}):=
{\frac {E_2^{a_1}
E_{12}^{a_2}
((3)_q^{-1}!E_{11212})^{a_3}
((2)_q^{-1}E_{112})^{a_4}
((3)_q^{-1}!E_{1112})^{a_5}
E_1^{a_6}}
{(a_1)_{q^3}!(a_2)_q!(a_3)_{q^3}!(a_4)_q!(a_5)_{q^3}!(a_6)_q!}
}, \\
Q_2({\bf a}):={\hat{\defacE}}_2^{a_1}{\hat{\defacE}}_{12}^{a_2}{\hat{\defacE}}_{11212}^{a_3}{\hat{\defacE}}_{112}^{a_4}
{\hat{\defacE}}_{1112}^{a_5}{\hat{\defacE}}_1^{a_6}.
\end{array}
\end{equation*}
Then we have
\begin{equation}\label{eqn:dualGtwo}
\begin{array}{l}
\vartheta^{\bhm,\defPi}(Q_1({\bf a}),\defUpsilon^{\bhmop,\defPi}(Q_2({\bf b})))
=\delta_{{\bf a},{\bf b}}\quad({\bf a},\,{\bf b}\in\bZ_{\geq 0}^6).
\end{array}
\end{equation}
By \eqref{eqn:dualGtwo},
$\{Q_1({\bf a})|{\bf a}\in\bZ_{\geq 0}^6\}$ and $\{Q_2({\bf a})|{\bf a}\in\bZ_{\geq 0}^6\}$
are ${\mathbb{K}}$-bases of ${\defU(\bhm,\defPi)}}%{{\defU(\bhm,\Pi)}$.
Let ${\acute{{\mathbb{A}}}}$ be the ${\mathbb{Z}}$-subalgebra of ${\mathbb{K}}$ generated by $q^{\pm 1}$
and $a^{\pm 1}$, i.e., ${\acute{{\mathbb{A}}}}={\mathbb{Z}}[q^{\pm 1},a^{\pm 1}]$.
For $t\in{\mathbb{J}}_{1,2}$, let ${\mathfrak{g}}_t$ be the ${\acute{{\mathbb{A}}}}$-submodules
of $\defU^{+,\flat}$ with the ${\acute{{\mathbb{A}}}}$-bases
$\{Q_t({\bf a})K_\lambda|{\bf a}\in\bZ_{\geq 0}^6,\,\lambda\in\defV_\bZ\}$.
Clearly ${\mathfrak{g}}_2$ is a ${\acute{{\mathbb{A}}}}$-subalgebra of $U$
with $\HopfD({\mathfrak{g}}_2)\subset{\mathfrak{g}}_2\otimes_{{\acute{{\mathbb{A}}}}}{\mathfrak{g}}_2$.
By \eqref{eqn:bprtoftB} and \eqref{eqn:dualGtwo}, we see the following.
\begin{equation}\label{eqn:Key}
\begin{array}{l}
\mbox{As a ${\acute{{\mathbb{A}}}}$-subalgebra of $U$, ${\mathfrak{g}}_1$ is generated by
${\frac {E_i^x} {(x)_{q_{ii}}!}}$} \\
\mbox{($i\inI(={\mathbb{J}}_{1,2})$, $x\in\bZ_{\geq 0}$) and $K_\lambda$ ($\lambda\in\defV_\bZ$).} \\
\mbox{${\mathfrak{g}}_1$ is a Hopf ${\acute{{\mathbb{A}}}}$-subalgebra of $U$.}
\end{array}
\end{equation} Then ${\mathfrak{g}}_2$ is also a Hopf ${\acute{{\mathbb{A}}}}$-algebra.

Let $n:=(1,2,1,2,1,2)\inI^{\defell(\twoh)}$.
Then we have
\begin{equation}\label{eqn:Keyeqa}
\begin{array}{ll}
{\bar{\defacE}}_{n;1}={\bar{\defacE}}_1, &  {\bar{\defacF}}_{n;1}={\bar{\defacF}}_1(=\defUpsilon^{\bhmop,\defPi}({\bar{\defacE}}_{n;1})), \\
{\bar{\defacE}}_{n;2}={\frac{{\dot{q}}_{12}^{-3}} {(3)_q!\Theta(q-1)^3\Theta(q^3-1)}}E_{1112},    &
{\bar{\defacF}}_{n;2}={\dot{q}}_{11}^3\defUpsilon^{\bhmop,\defPi}({\bar{\defacE}}_{n;2}), \\
{\bar{\defacE}}_{n;3}={\frac{{\dot{q}}_{12}^{-2}} {(2)_q!\Theta(q^3-1)\Theta(q-1)^2}}E_{112}, &
{\bar{\defacF}}_{n;3}={\dot{q}}_{11}^4\defUpsilon^{\bhmop,\defPi}({\bar{\defacE}}_{n;3}), \\
{\bar{\defacE}}_{n;4}={\frac{{\dot{q}}_{12}^{-4}} {(3)_q!\Theta(q^3-1)^2\Theta(q-1)^3}}E_{11212}, &
{\bar{\defacF}}_{n;4}={\dot{q}}_{11}^6\defUpsilon^{\bhmop,\defPi}({\bar{\defacE}}_{n;4}), \\
{\bar{\defacE}}_{n;5}={\frac{{\dot{q}}_{12}^{-1}} {\Theta(q-1)\Theta(q^3-1)}}E_{12}, &
{\bar{\defacF}}_{n;5}={\dot{q}}_{11}^3\defUpsilon^{\bhmop,\defPi}({\bar{\defacE}}_{n;5}), \\
{\bar{\defacE}}_{n;6}={\bar{\defacE}}_2, & {\bar{\defacF}}_{n;6}={\bar{\defacF}}_2(=\defUpsilon^{\bhmop,\defPi}({\bar{\defacE}}_{n;6})),
\end{array}
\end{equation} where recall that ${\dot{q}}_{ij}^2=a$.
Let $n^\prime:=(2,1,2,1,2,1)\inI^{\defell(\twoh)}$. By \eqref{eqn:InvLusTi} and  \eqref{eqn:Enlwo}, we have
\begin{equation}\label{eqn:Keyeqb}
{\bar{\defacE}}_{n^\prime;t}=\defGamma^{\bhm^{\mathrm{op}},\defPi}({\bar{\defacE}}_{n;7-t}),\,\,{\bar{\defacF}}_{n^\prime;t}=\defGamma^{\bhm^{\mathrm{op}},\defPi}({\bar{\defacF}}_{n;7-t})
\quad (t\in{\mathbb{J}}_{1,6}).
\end{equation}

Let ${\bar {\mathfrak{g}}}^+$ be the ${\acute{{\mathbb{A}}}}$-subalgebra of $U$ generated by
${\frac {{\bar{\defacE}}_i^x} {(x)_{q_{ii}}!}}$ ($i\inI(={\mathbb{J}}_{1,2})$, $x\in\bZ_{\geq 0}$) and $K_\lambda$ ($\lambda\in\defV_\bZ$).
Let $y_{2t-1}:=q$, $y_{2t}:=q^3$ for $t\in{\mathbb{J}}_{1,3}$.
Let $y^\prime_t:=y_{7-t}$  for $t\in{\mathbb{J}}_{1,6}$.
By \eqref{eqn:EnyEnz}, \eqref{eqn:Key}, \eqref{eqn:Keyeqa} and \eqref{eqn:Keyeqb}, we see the following.
\begin{equation}\label{eqn:BdefEKey}
\begin{array}{l}
\mbox{For a bijection $\sigma:{\mathbb{J}}_{1,6}\to{\mathbb{J}}_{1,6}$, the elements} \\
\mbox{${\frac {{\bar{\defacE}}_{n;\sigma(1)}^{x_1}} {(x_1)_{y_{\sigma(1)}}!}}\cdots{\frac {{\bar{\defacE}}_{n;\sigma(6)}^{x_6}} {(x_6)_{y_{\sigma(6)}}!}}K_\lambda$
(resp. ${\frac {{\bar{\defacE}}_{n^\prime;\sigma(1)}^{x_1}} {(x_1)_{y^\prime_{\sigma(1)}}!}}\cdots{\frac {{\bar{\defacE}}_{n^\prime;\sigma(6)}^{x_6}} {(x_6)_{y^\prime_{\sigma(6)}}!}}K_\lambda$)} \\
\mbox{($x_t\in\bZ_{\geq 0}$ ($t\in{\mathbb{J}}_{1,6}$), $\lambda\in\defV_\bZ$) form a ${\acute{{\mathbb{A}}}}$-base of ${\bar {\mathfrak{g}}}^+$.} \\
\mbox{${\bar {\mathfrak{g}}}^+$ is a Hopf ${\acute{{\mathbb{A}}}}$-subalgebra of $U$.}
\end{array}
\end{equation}

Let ${\bar {\mathfrak{g}}}^-$ be the ${\acute{{\mathbb{A}}}}$-subalgebra of $U$ generated by
${\frac {{\bar{\defacF}}_i^x} {(x)_{q_{ii}}!}}$ ($i\inI(={\mathbb{J}}_{1,2})$, $x\in\bZ_{\geq 0}$) and $L_\lambda$ ($\lambda\in\defV_\bZ$).
By \eqref{eqn:Keyeqa}, \eqref{eqn:Keyeqb}
and \eqref{eqn:BdefEKey}, we see the following.
\begin{equation}\label{eqn:BdefFKey}
\begin{array}{l}
\mbox{For a bijection $\sigma:{\mathbb{J}}_{1,6}\to{\mathbb{J}}_{1,6}$, the elements} \\
\mbox{${\frac {{\bar{\defacF}}_{n;\sigma(1)}^{x_1}} {(x_1)_{y_{\sigma(1)}}!}}\cdots{\frac {{\bar{\defacF}}_{n;\sigma(6)}^{x_6}} {(x_6)_{y_{\sigma(6)}}!}}L_\lambda$
(resp. ${\frac {{\bar{\defacF}}_{n^\prime;\sigma(1)}^{x_1}} {(x_1)_{y^\prime_{\sigma(1)}}!}}\cdots{\frac {{\bar{\defacF}}_{n^\prime;\sigma(6)}^{x_6}} {(x_6)_{y^\prime_{\sigma(6)}}!}}L_\lambda$)} \\
\mbox{($x_t\in\bZ_{\geq 0}$ ($t\in{\mathbb{J}}_{1,6}$), $\lambda\in\defV_\bZ$) form a ${\acute{{\mathbb{A}}}}$-base of ${\bar {\mathfrak{g}}}^-$.} \\
\mbox{${\bar {\mathfrak{g}}}^-$ is a Hopf ${\acute{{\mathbb{A}}}}$-subalgebra of $U$.}
\end{array}
\end{equation}

\subsection{General case}
Let ${\mathbb{A}}$ be the ${\mathbb{Z}}$-subalgebra of ${\mathbb{K}}$ generated by
${\dot{q}}_{ij}^{\pm 1}$ for all $i$, $j\inI$.
$n=(n_1,\ldots,n_{\defell(\twoh)})\inI^{\defell(\twoh)}$ be as in Subsection~\ref{subsection:somno}.
Let $\pi}%{\Pi_0:=\pi}%{\Pi$ and
$\pi}%{\Pi_t:=\tau^\bhm_{n_t}\cdots\tau^\bhm_{n_1}\pi}%{\Pi$
($t\in{\mathbb{J}}_{1,\defell(\twoh)}$).
For $t\in{\mathbb{J}}_{1,\kappa}$ and $x\in\bZ_{\geq 0}$, let
\begin{equation*}
{\bar{\defacE}}_{n;t}^{(x)}:={\frac {{\bar{\defacE}}_{n;t}^x}
{(x)_{\chi(\pi}%{\Pi_{t-1}(n_t),\pi}%{\Pi_{t-1}(n_t))}!}},\quad
{\bar{\defacF}}_{n;t}^{(x)}:={\frac {{\bar{\defacF}}_{n;t}^x}
{(x)_{\chi(\pi}%{\Pi_{t-1}(n_t),\pi}%{\Pi_{t-1}(n_t))}!}}.
\end{equation*}
Let $\defUp_\bA$ (resp. $\defUm_\bA$) be the ${\mathbb{A}}$-subalgebra of $\defU^+$
(resp. $\defU^-$) generated by
${\frac {{\bar{\defacE}}_i^x} {(x)_{q_{ii}}!}}$
(resp. ${\frac {{\bar{\defacF}}_i^x} {(x)_{q_{ii}}!}}$)
with $i\inI$ and $x\in\bZ_{\geq 0}$.
\begin{theorem}
Let $\sigma:{\mathbb{J}}_{1,\defell(\twoh)}\to{\mathbb{J}}_{1,\defell(\twoh)}$ be a bijection.
\newline\newline
{\rm{(1)}}
The elements
\begin{equation} \label{eqn:mainUpA}
\mbox{${\bar{\defacE}}_{n;\sigma(1)}^{(x_1)}\cdots{\bar{\defacE}}_{n:\sigma(\defell(\twoh))}^{(x_{\defell(\twoh)})}$
with $x_t\in\bZ_{\geq 0}$ {\rm{(}}$t\in{\mathbb{J}}_{1,\defell(\twoh)}${\rm{)}} }
\end{equation} form an
${\mathbb{A}}$-base for $\defUp_\bA$.
\newline
{\rm{(2)}}
The elements
\begin{equation} \label{eqn:mainUmA}
\mbox{${\bar{\defacF}}_{n;\sigma(1)}^{(x_1)}\cdots{\bar{\defacF}}_{n:\sigma(\defell(\twoh))}^{(x_{\defell(\twoh)})}$
with $x_t\in\bZ_{\geq 0}$ {\rm{(}}$t\in{\mathbb{J}}_{1,\defell(\twoh)}${\rm{)}} }
\end{equation} form an
${\mathbb{A}}$-base for $\defUm_\bA$.
\end{theorem}
{\it{Proof.}} We only prove (1), as (2) can be proved similarly or obtained from (1) via Chevalley isomorphism.
If $|I|=1$, the claim is clear.

If $|I|=2$ and $A$ is of type $G_2$, the result is obtained in \eqref{eqn:BdefEKey} and \eqref{eqn:BdefFKey}.
For other rank two types, the claim can be proved in a similar and in fact easier way.

Let us consider the higher rank cases.
Let $Z$ be the free ${\mathbb{A}}$-submodule of $\defU^+$ with the
free ${\mathbb{A}}$-basis formed by the elements of \eqref{eqn:mainUpA} (for $\sigma:={\mathrm{id}}_{{\mathbb{J}}_{1,\defell(\twoh)}}$,
see Theorem~\ref{theorem:LusPBW}~(1)).
By an argument similar to \cite[Proposition~8.20]{Jan96}
and by the claim for rank-two cases, we see that
${\bar{\defacE}}_{n;t}^{(x)}\in\defUp_\bA$, so $Z\subseteq\defUp_\bA$.
Now by \eqref{eqn:prtwoh}, we have $Z=\defUp_\bA$.
Thus the claim follows from \eqref{eqn:EnyEnz}. \hfill $\Box$
\newline\par
For $k$, $r\in\bZ_{\geq 0}$ and $x\in{\bK^\times}$
with $(k+r)_x!\ne 0$,
let ${{k} \choose {r}}_x:={\frac {(k+r)_x!} {(k)_x!(r)_x!}}$.
For $x\in{\bK^\times}$ with $x^b\ne 1$ for all $b\in{\mathbb{N}}$ and for $X$, $Y\in\defU^0$, $l\in{\mathbb{Z}}$ and $p\in\bZ_{\geq 0}$, let
\begin{equation*}
\left[\begin{array}{c} X,Y,l \\ p\end{array}\right]_x
:=\prod_{t=1}^p{\frac {x^{l-t+1}X-Y} {x^t-1}}.
\end{equation*} Then we have
\begin{equation*}
\begin{array}{l}
\left[\begin{array}{c} X,Y,l \\ p\end{array}\right]_x
-\left[\begin{array}{c} X,Y,l+1 \\ p\end{array}\right]_x
=-x^{l-p+1}\left[\begin{array}{c} X,Y,l \\ p-1\end{array}\right]_xX, \\
\left[\begin{array}{c} X,Y,0 \\ l\end{array}\right]_x
\left[\begin{array}{c} X,Y,-l \\ p\end{array}\right]_x
={{p+l} \choose {p}}_x\left[\begin{array}{c} X,Y,0 \\ p+l\end{array}\right]_x.
\end{array}
\end{equation*}
Let $\defUo_\bA$ be the ${\mathbb{A}}$-subalgebra generated by
$K_\lambdaL_\mu$ ($\lambda$, $\mu\in\defV_\bZ$)
and $\left[\begin{array}{c} K_{\pi}%{\Pi(i)},L_{\pi}%{\Pi(i)},l \\ p\end{array}\right]_{q_{ii}}$
($i\inI$, $l\in{\mathbb{Z}}$, $p\in\bZ_{\geq 0}$).
By a standard argument, we have the following two lemmas.
\begin{lemma} \label{lemma:bsUoA} The elements
\begin{equation*}
\prod_{i\inI}K_{\pi}%{\Pi(i)}^{x_i}(K_{\pi}%{\Pi(i)}L_{\pi}%{\Pi(i)})^{y_i}
\left[\begin{array}{c} K_{\pi}%{\Pi(i)},L_{\pi}%{\Pi(i)},0
\\ z_i\end{array}\right]_{q_{ii}}\quad
(x_i\in{\mathbb{Z}}_{0,1},\,y_i\in{\mathbb{Z}},\,z_i\in\bZ_{\geq 0})
\end{equation*} form an ${\mathbb{A}}$-base of $\defUo_\bA$.
\end{lemma}

Let $\defU_\bA$ be the ${\mathbb{A}}$-subalgebra of $U$ generated by
$\defUp_\bA$ and $\defUm_\bA$.

\begin{lemma} \label{lemma:triUA}
We have $\defUo_\bA\subset\defU_\bA$, and we have
the ${\mathbb{A}}$-module isomorphism
\begin{equation*}
{\mathbf{m}}_{\mathbb{A}}:\defUm_\bA\otimes_{\mathbb{A}}\defUo_\bA\otimes_{\mathbb{A}}\defUp_\bA\longrightarrow \defU_\bA
\end{equation*} given by ${\mathbf{m}}_{\mathbb{A}}(X\otimes Y\otimes Z):=XYZ$.
\end{lemma}

\begin{center}
{\bf{Acknowledgments}}
\end{center}

We thank the referee for careful reading and kind comments. The third author would like to express his heartfelt thanks to Professor Naihong~Hu for valuable communication.

redpajama_set_name: RedPajamaArXiv


surrounding the impressive new school buildings.
The majority of the work undertaken by Bethell consisted of the installation of bespoke high quality paving.
The site was challenging as it is steeply sloped and paving was constructed on a variety of levels overlooking Sheffield Town Centre and each 'terrace' was constructed differently using a variety of construction materials.
Other works included construction of a MUGA pitch, substantial feature gabion walls, bulk earthworks and all asphalting works.


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text:
from __future__ import print_function,division
from six import iteritems,iterkeys

import random

class RandomDict(object):
    def __init__(self): # O(1)
        self.dictionary = {}
        self.indexdict = {}
        self.next_index = 0
        self.removed_indices = None
        self.len = 0

    def __len__(self): # might as well include this
        return self.len

    def __contains__(self, key):
        return key in self.dictionary

    def __getitem__(self, key): # O(1)
        return self.dictionary[key][1]

    def __setitem__(self, key, value): # O(1)
        if key in self.dictionary: # O(1)
            self.dictionary[key][1] = value # O(1)
            return
        if self.removed_indices is None:
            index = self.next_index
            self.next_index += 1
        else:
            index = self.removed_indices[0]
            self.removed_indices = self.removed_indices[1]
        self.dictionary[key] = [index, value] # O(1)
        self.indexdict[index] = key # O(1)
        self.len += 1

    def get(self, key, defaultValue): # O(1)
        v = self.dictionary.get(key,None)
        if v == None: return defaultValue
        return v[1]

    def setdefault(self,key,defaultValue):
        if key in self.dictionary: # O(1)
            return self.dictionary[key][1] # O(1)
        if self.removed_indices is None:
            index = self.next_index
            self.next_index += 1
        else:
            index = self.removed_indices[0]
            self.removed_indices = self.removed_indices[1]
        self.dictionary[key] = [index, defaultValue] # O(1)
        self.indexdict[index] = key # O(1)
        self.len += 1
        return self.dictionary[key][1]

    def __delitem__(self, key): # O(1)
        index = self.dictionary[key][0] # O(1)
        del self.dictionary[key] # O(1)
        del self.indexdict[index] # O(1)
        self.removed_indices = (index, self.removed_indices)
        self.len -= 1

    def random_key(self,weight=None):
        """Randomly select a key in the dictionary.  Unweighted
        version is O(log(next_index/len)).  Weighted version selects
        items proportionally to weight(key,value), and runs in time
        O(len)
        """
        if self.len == 0: # which is usually close to O(1)
            raise KeyError
        if weight==None:
            while True:
                r = random.randrange(0, self.next_index)
                if r in self.indexdict:
                    return self.indexdict[r]
        else:
            weights = {}
            sumweight = 0.0
            for k,v in iteritems(self.dictionary):
                w = weight(k,v)
                weights[k] = w
                sumweight += weights[k]
            u = random.random()*sumweight
            for k,v in iteritems(self.dictionary):
                u -= weights[k]
                if u <= 0: return k
            print("Numerical error in random_key")
            return iterkeys(self.dictionary).next()

redpajama_set_name: RedPajamaGithub


text:
Q: After deletion of chrome extension remove some html elements from website Hi I am new to chrome extension. I have build the basic chrome extension and I want to install it using inline installation. I followed the following  link :
https://developer.chrome.com/webstore/inline_installation#already-installed
All is working properly. I want to check if extension is already installed or not so I referred the above document, whenever user install the extension at that time we are appending the new div using content script to my installation page on website. If that div is present it means the extension is already installed.
Following code is added in my content_script of chrome extension for appending the div whenever the extension is installed:
var isInstalledNode = document.createElement('div');
isInstalledNode.id = 'extension-is-installed';
document.body.appendChild(isInstalledNode);

Then my installation page can check for the presence of that DOM node, which signals that the extension is already installed:
if (document.getElementById('extension-is-installed')) {
  document.getElementById('install-button').style.display = 'none';
}

But I am facing one problem, whenever I deleted my extension from settings/extensions, still the div is present on extension's installation page.
Is their any provision to remove the div when my extension is deleted or removed from browser?

A: You can't catch the uninstallation event for your own extension, though there is an management.onUninstalled, it is used for listening to other extensions/apps. See following answer for more details:
How to listen to uninstallation event in javascript from a chrome extension?
I just come up with two workarounds, one is creating another extension to listen to the uninstall of your first extension; another one is using runtime.setUninstallURL to visit a URL upon uninstallation, you can do something in server if you want.
BTW, for isInstalledNode, you needn't/shouldn't set the id, since there may be other elements in the webpage with the same id, you just need to create the node and use document.body.contains(isIntalledNode) to check if it exists.


text:
Q: Walking through rows within CSV I am creating something whereby you can use a next and previous button to walk through different rows within a CSV.  The first step of my process is to retrieve all of the rows within the CSV file.  To do this I have created the following function.
function getRows() {
    var regex = /^([a-zA-Z0-9\s\S_\\.\-:])+(.csv|.txt)$/;
    if (regex.test($("#fileOne").val().toLowerCase())) {
        if (typeof (FileReader) != "undefined") {
            var reader = new FileReader();
            reader.onload = function (e) {
                return e.target.result.split("\n");
            };
            reader.readAsText($("#fileOne")[0].files[0]);
        } else {
            alert("This browser does not support HTML5.");
        }
    } else {
        alert("Please upload a valid CSV file.");
    }
}

When the preview button is pressed, I am currently doing the following
$("#preview-btn").bind("click", function () {
    var rows = getRows();
    console.log(rows);
});

At the moment it is printing undefined to the console.  The idea is that I display the first row in the preview-btn event, and then use a counter to display different rows in the next and previous button events.  At the moment though I am trying to figure out why the rows are not being returned to me but instead I get undefined.
Am I missing something when I return the rows?
Thanks


We saw 'What Is WAMP And WAMPServer' in the last post, here we are going to learn about 'LAMP' which is nothing but a substitute to WAMP when one has hosted his website on 'Linux Hosting Server'. LAMP consists of Linux, Apache, MySQL and PHP/Perl/Python, where Linux is an operating system, Apache is a Web Server, MySQL fulfills database requirements and PHP/Perl/Python work as scripting languages. The idea of LAMP was originally invented in 1999 by 'Michael Kunze' in a German magazine, though the four elements were not approached to work together but in the later period these open source applications were made available for free of cost and as a bunch called as 'LAMP stack'. Because of having the ability to make a strong web application platform, LAMP gained the popularity in very early stage in such a way that in today's market LAMP services are available as by default with almost all the website hosting services providers.
There are multiple security options available while using LAMP, you can configure Apache and PHP to remain hidden so that your information would not be leaked by any possible hacking attacks. You can find more about LAMP security at :- Is Your 'LAMP' Secure Enough?. LAMP tools are easily available with the features of easy configuration and being very robust. Being as an open-source application, LAMP is supported by the larger community, which constantly adds features and development to meet the requirements of it's users. For deployment, configuration and maintenance of LAMP, one needs to spend very little effort and time, furthermore you don't need any type of administrative details for it.


Jean-Nicolas Gérard was born in Brazzaville (Congo) in 1954 and returned to France in 1961. He started studying ceramics in 1978 and was Jean Biagini's student at École des Beaux-Arts in Aix-en-Provence. He also trained with Claire Bogino. Often labelled the potters' potter Gerard's work has a spontaneity that so many strive for.
Gérard's work has now gained international acclaim and he has exhibited all over the world, including America, Australia, China and Japan. He is one of those rare potters who brings genuine life and gusto to contemporary slipware, investing the tradition of terre vernissée with a fresh and expressive energy unlike any other.