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MSC 2010: Associative rings and algebras
16Txx
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48 results in 16Txx
A categorical approach to dynamical quantum groups
Lie algebras and Lie superalgebras
Hopf algebras, quantum groups and related topics
Artem Kalmykov, Pavel Safronov
Journal: Forum of Mathematics, Sigma / Volume 10 / 2022
Published online by Cambridge University Press: 02 September 2022, e76
We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.
Quasi-solvable lattice models for $\operatorname {Sp}_{2n}$ and $\operatorname {SO}_{2n+1}$ Demazure atoms and characters
Algebraic combinatorics
Equilibrium statistical mechanics
Valentin Buciumas, Travis Scrimshaw
Published online by Cambridge University Press: 18 July 2022, e53
We construct coloured lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang–Baxter equation that allow us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.
On the computation of Hopf 2-cocycles with an example of diagonal type
Homological methods
Agustín García Iglesias, José Ignacio Sánchez
Journal: Glasgow Mathematical Journal / Volume 65 / Issue 1 / January 2023
We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.
Corrigendum: Liftings of Jordan and Super Jordan Planes
Nicolás Andruskiewitsch, Iván Angiono, István Heckenberger
Journal: Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 2 / May 2022
We complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in [N. Andruskiewitsch, I. Angiono and I. Heckenberger, Liftings of Jordan and super Jordan planes, Proc. Edinb. Math. Soc., II. Ser. 61(3) (2018), 661–672.].
Specht property for the algebra of upper triangular matrices of size two with a Taft's algebra action
Rings with polynomial identity
Lucio Centrone, Alejandro Estrada
Journal: Canadian Mathematical Bulletin , First View
Published online by Cambridge University Press: 16 May 2022, pp. 1-21
Let F be a field of characteristic zero, and let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft's algebras $H_m$ on $UT_2$ and its $H_m$ -identities. In this paper, we give a complete description of the space of multilinear $H_m$ -identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of $H_m$ -module algebras generated by $UT_2$ has the Specht property, i.e., every $T^{H_m}$ -ideal containing the $H_m$ -identities of $UT_2$ is finitely based.
CATEGORICAL AND K-THEORETIC HALL ALGEBRAS FOR QUIVERS WITH POTENTIAL
Projective and enumerative geometry
Tudor Pădurariu
Journal: Journal of the Institute of Mathematics of Jussieu , First View
Given a quiver with potential $(Q,W)$ , Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of $(Q,W)$ . Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair $(\widetilde{Q}, \widetilde{W})$ whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian $Y_{\text {MO}}(\mathfrak {g}_{Q})$ .
For a quiver with potential $(Q,W)$ , we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for $(\widetilde{Q}, \widetilde{W})$ to recover the positive part of quantum affine algebra $U_{q}(\widehat {\mathfrak {g}_{Q}})$ defined by Okounkov–Smirnov.
An operadic approach to substitution in Lie–Butcher series
Ludwig Rahm
Published online by Cambridge University Press: 31 March 2022, e20
The paper follows an operadic approach to provide a bialgebraic description of substitution for Lie–Butcher series. We first show how the well-known bialgebraic description for substitution in Butcher's B-series can be obtained from the pre-Lie operad. We then apply the same construction to the post-Lie operad to arrive at a bialgebra $\mathcal {Q}$ . By considering a module over the post-Lie operad, we get a cointeraction between $\mathcal {Q}$ and the Hopf algebra $\mathcal {H}_{N}$ that describes composition for Lie–Butcher series. We use this coaction to describe substitution for Lie–Butcher series.
The finite dual of commutative-by-finite Hopf algebras
Chain conditions, growth conditions, and other forms of finiteness
K. A. Brown, M. Couto, A. Jahn
Published online by Cambridge University Press: 21 March 2022, pp. 62-89
The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra H is studied. Such a Hopf algebra H is an extension of an affine commutative Hopf algebra A by a finite dimensional Hopf algebra $\overline{H}$ . The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $\overline{H}^{\ast}$ by the finite dual $A^{\circ}$ of A. This decomposition is then further analysed using the Cartier–Gabriel–Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$ . The detailed consequences for a number of families of examples are then studied.
Resonance-based schemes for dispersive equations via decorated trees
Equations of mathematical physics and other areas of application
Partial differential equations, initial value and time-dependent initial-boundary value problems
Yvain Bruned, Katharina Schratz
Journal: Forum of Mathematics, Pi / Volume 10 / 2022
Published online by Cambridge University Press: 13 January 2022, e2
We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.
HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS
General and miscellaneous
CRIS NEGRON, JULIA PEVTSOVA
Journal: Nagoya Mathematical Journal / Volume 247 / September 2022
We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz' support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.
Defining relations of quantum symmetric pair coideal subalgebras
Global differential geometry
Stefan Kolb, Milen Yakimov
Journal: Forum of Mathematics, Sigma / Volume 9 / 2021
Published online by Cambridge University Press: 05 October 2021, e67
We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$-graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.
On holomorphic reflexivity conditions for complex Lie groups
Special aspects of infinite or finite groups
O. Yu. Aristov
Journal: Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 4 / November 2021
Published online by Cambridge University Press: 30 September 2021, pp. 800-821
We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$ .
Wick polynomials in noncommutative probability: a group-theoretical approach
Selfadjoint operator algebras
General nonassociative rings
Kurusch Ebrahimi-Fard, Frédéric Patras, Nikolas Tapia, Lorenzo Zambotti
Journal: Canadian Journal of Mathematics / Volume 74 / Issue 6 / December 2022
Published online by Cambridge University Press: 25 August 2021, pp. 1673-1699
Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.
Two theorems on balanced braces
Rings and algebras with additional structure
Wolfgang Rump
Published online by Cambridge University Press: 05 May 2021, pp. 262-278
Two theorems of Gateva-Ivanova [Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math. 338 (2018), 649–701] on square-free set-theoretic solutions to the Yang–Baxter equation are extended to a wide class of solutions. The square-free hypothesis is almost completely removed. Gateva-Ivanova and Majid's 'cyclic' condition ${\boldsymbol {\rm lri}}$ is shown to be equivalent to balancedness, introduced in Rump [A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005), 40–55]. Basic results on balanced solutions are established. For example, it is proved that every finite, not necessarily square-free, balanced brace determines a multipermutation solution.
Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups
Groups and algebras in quantum theory
Groupoids
Journal: Canadian Mathematical Bulletin / Volume 65 / Issue 1 / March 2022
Published online by Cambridge University Press: 20 April 2021, pp. 225-233
Set-theoretic solutions to the Yang–Baxter equation can be classified by their universal coverings and their fundamental groupoids. Extending previous results, universal coverings of irreducible involutive solutions are classified in the degenerate case. These solutions are described in terms of a group with a distinguished self-map. The classification in the nondegenerate case is simplified and compared with the description in the degenerate case.
Centrification of algebras and Hopf algebras
Rings and algebras arising under various constructions
Dmitriy Rumynin, Matthew Westaway
We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.
Depth-graded motivic multiple zeta values
Zeta and $L$-functions: analytic theory
Ring extensions and related topics
Francis Brown
Journal: Compositio Mathematica / Volume 157 / Issue 3 / March 2021
Published online by Cambridge University Press: 22 March 2021, pp. 529-572
We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.
McKay matrices for finite-dimensional Hopf algebras
Grothendieck groups and $K_0$
Georgia Benkart, Rekha Biswal, Ellen Kirkman, Van C. Nguyen, Jieru Zhu
Journal: Canadian Journal of Mathematics / Volume 74 / Issue 3 / June 2022
Published online by Cambridge University Press: 08 February 2021, pp. 686-731
For a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.
EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2
NICOLÁS ANDRUSKIEWITSCH, DIRCEU BAGIO, SARADIA DELLA FLORA, DAIANA FLÔRES
Published online by Cambridge University Press: 01 December 2020, pp. 65-78
We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.
On the adjoint representation of a hopf algebra
Stefan Kolb, Martin Lorenz, Bach Nguyen, Ramy Yammine
Published online by Cambridge University Press: 11 November 2020, pp. 1092-1099
We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$ , defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$ . This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.