SL_q(2,R) at Roots of Unity

A new aspect of representations of U_q(\hat sl_2) - root of unity case

The structure of the tensor product representation v_{λ_1}(x)\otimes V_{λ_2}(y) of U_q(\hat sl_2) is investigated at roots of unity. A polynomial identity is derived as an outcome. Also, new bases of v_{λ_1}(x)\otimes V_{λ_2}(y) are established under certain conditions.

Quantum Weyl algebras and reflection equation algebras at a root of unity

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].

A[Sl_q(2)] at roots of unity is a free module over A[Sl(2)]

It is shown that when q is a primitive root of unity of order not equal to 2 mod 4, A(SL_q(2)) is a free module of finite rank over the coordinate ring of the classical group SL(2). An explicit set of generators is provided.

Two Dimensional Fractional Supersymmetry from the Quantum Poincare Group at Roots of Unity

A group theoretical understanding of the two dimensional fractional supersymmetry is given in terms of the quantum Poincare group at roots of unity. The fractional supersymmetry algebra and the quantum group dual to it are presented and the pseudo-unitary, irreducible representations of them are obtained. The matrix elements of these representations are explicitly constructed.

$\mathfrak{R}$-matrix for quantum superalgebra $\mathfrak{sl}(2|1)$ at roots of unity and its application to centralizer algebras

We consider fundamental facts from the theory of Hopf superalgebras. We use them to construct the quantum double of the quantum superalgebra $sl(2|1)$ at roots of unity. Thus we obtain a multiplicative formula for universal $R$-matrix. Next we construct an $R$-matrix to investigate parametrized family of centralizer algebras. We give multiplication laws in particular case and describe a structure of such algebras in the general case.

Zelevinsky's involution at roots of unity

We give a combinatorial algorithm for computing Zelevinsky's involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra $\H_m(t)$ when $t$ is a primitive $n$th root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of $\Zb/n\Zb$-graded vector spaces.

Quantum affine algebras at roots of unity and equivariant K-theory

In this short note, we show that the Ginzburg-Vasserot map between the quantum affine algebra of type A_(n-1) and the equivariant K-theory group of the Steinberg Variety (of n-step flags in C^d) restricts and remains surjective at the level of the integral forms. In particular, this shows that the parametrization of irreducible, finite-dimensional modules and the Kazhdan-Lusztig multiplicity formulas are valid for the (restricted) specialization at roots of unity.

The t-Analogs of q-Characters at Roots of Unity for Quantum Affine Algebras and Beyound

The q-characters were introduced by Frenkel and mReshetikhin to study finite dimensional representations of the untwisted quantum affine algebra for q generic. The $ε$-characters at roots of unity were constructed by Frenkel and Mukhin to study finite dimensional representations of various specializations of the quantum affine algebra at $q^s=1$. In the finite simply laced case Nakajima defined deformations of q-characters called q,t-characters, for q generic and also at roots of unity. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties…

Braiding for the quantum gl_2 at roots of unity

In our preceding papers we started considering the categories of tangles with flat G-connections in their complements, where G is a simple complex algebraic group. The braiding (or the commutativity constraint) in such categories satisfies the holonomy Yang-Baxter equation and it is this property which is essential for our construction of invariants of tangles with flat G-connections in their complements. In this paper, to any pair of irreducible modules over the quantized universal enveloping algebra of gl_2 at a root of unity, we associate a solution of the holonomy Yang-Baxter equation.

Irreducible Modules of Finite Dimensional Quantum Algebras of type A at Roots of Unity

Specializing properly the parameters contained in the maximal cyclic representation of the non-restricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. We show that the submodule generated by the primitive vector can be identified with an irreducble highest weight module of the finite dimensional A-type quantum algebra which is defined as the subalgebra of the restricted quantum algebra at roots of unity.

Quantum Groups at Roots of Unity and Modularity

For each compact, simple, simply-connected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine when at fractional levels the corresponding category is modular. We also give a quantum version of the Racah formula for the decomposition of the tensor product. This work relies on developing the basic representation theory of quantum groups at roots of unity, including Harish-Chandra's Theorem. It generalizes previous work which applied only to fractional …

单位根下量子化Weyl代数上的单模

Simple Modules Over Quantized Weyl Algebras At Roots Of Unity

该研究完成了代数闭域上秩二单位根量子化Weyl代数的单模分类工作。

Irreducible representations of q-Schur superalgebras at a root of unity

Under the assumption that the quantum parameter $q$ is an $l$-th primitive root of unity with $l$ odd in a field $F$ of characteristic 0 and $m+n\geq r$, we obtained a complete classification of irreducible modules of the $q$-Schur superalgebra introduced H. Rui and the first Author.

Representations at a Root of Unity of q-Oscillators and Quantum Kac-Moody algebras

Here is a list of chapters: 1 Introduction 2 Notation and preliminaries Part I: Finite quantum groups 3 2x2 Matrix quantum groups and the quantum plane 4 Quantum enveloping algebras at a root of unity Part II: q-Oscillators 5 Representations of q-oscillators at a root of unity 6 qr-Oscillator at a root of unity Part III: Infinite quantum groups 7 Quantum affine algebras 8 Quantum affine algebras at a root of unity

Representations Of Quantum Spatial Ageing Algebra At Root Of Unity

In this article, the quantum spatial ageing algebra due to V. V. Bavula and T. Lu has been studied and a full classification of simple modules are given at root of unity

On quantized algebra of Wess-Zumino differential operators at roots of unity

Quantum algebra of differential operators are studied

Categorification at prime roots of unity and hopfological finiteness

We survey some recent results in hopfological algebra and the program of categorification at prime roots of unity. A categorical Jones-Wenzl projector at prime roots of unity is studied, and it is shown that this projector is hopfologically finite in special characteristics, while generically it is infinite.

Affine quantum Schur algebras at roots of unity

We will classify finite dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize \cite[(6.5f) and (6.5g)]{Gr80} to the affine case in this paper.

Ribbon tableaux, ribbon rigged configurations and Hall-Littlewood functions at roots of unity

Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using A. Schilling's bijection between ribbon tableaux and ribbon rigged configurations.

The center of a quantized enveloping algebra at an even root of unity

We will give an explicit description of the center of the De Concini-Kac type specialization of a quantized enveloping algebra at an even root of unity. The case of an odd root of unity was already dealt with by De Concini-Kac-Procesi. Our description in the even case is similar to but a little more complicated than the odd case.

A categorification of quantum sl(2) at prime roots of unity

We categorify the Beilinson-Lusztig-MacPherson integral form of quantum sl(2) specialized at a prime root of unity.

R-matrices for highest weight representations of sl_q(2,C) at roots of unity

The general formula for R-matrices of slq(2,C) for the highest weight repre- sentations both for general q and for q being a root of unity by generalizing G.Gomez's and G.Sierra's one for semiperiodic representations of slq(2,C) at roots of unity is presented.

Tensor Product of the Fundamental Representations for the Quantum Loop Algebras of Type A at Roots of Unity

In this paper, we consider the necessary and sufficient conditions for the tensor product of the fundamental representations for the restricted quantum loop algebras of type A at roots of unity to be irreducible.

Affine open covering of the quantized flag manifolds at roots of unity

We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of Rosenberg.

Inductive Construction of Nilpotent Modules of Quantum Groups at Roots of Unity

The purpose of this paper is to prove that we can construct all finite dimensional irreducible nilpotent modules of type 1 inductively by using Schnizer homomorphisms for quantum algebra at roots of unity of type A, B, C, D or G.

Quantum Dilogarithm Identities at Root of Unity

We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the non-compact quantum dilogarithm at $b=1$.

Nilpotent representations of classical quantum groups at roots of unity

Properly specializing the parameters in ``Schnizer modules'', for type A,B,C and D, we get its unique primitive vector. Then we show that the module generated by the primitive vector is an irreducible highest weight module of finite dimensional classical quantum groups at roots of unity.

The projective class ring of half-quantum groups at roots of unity

We describe the structure of the Grothendieck ring of projective modules of basic Hopf algebras using a positive integer determined by the composition series of the principal indecomposable projective module.

Dipper Donkin Quantized Matrix Algebra and Reflection Equation Algebra at root of unity

In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of rank $2$ along with a class of finite dimensional indecomposable modules are given.

Evaluation representations of quantum affine algebras at roots of unity

The purpose of this paper is to compute the Drinfel'd polynomials for two types of evaluation representations of quantum affine algebras at roots of unity and construct those representations as the submodules of evaluation Schnizer modules. Moreover, we obtain the necessary and sufficient condition for that the two types of evaluation representations are isomorphic to each other.

q-Ultraspherical polynomials for q a root of unity

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of $q$-beta integrals of Ramanujan.

Roots of Unity: Representations of Quantum Groups

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.

Some Realization of $gl_q(n)$-covariant Oscillator Algebra and $gl_q(n)$-covariant q-Virasoro Algebra with $q$ a root of unity

In this paper some realization of $gl_q(n)$-covariant oscillators is obtained when $q$ is a root of unity. And the $gl_q(n)$-covariant q-Virasoro algebra is presented by using the $gl_q(n)$-covariant oscillators.

A Frobenius homomorphism for Lusztig's quantum groups over arbitrary roots of unity

For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra. In this article we define and describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial …

The centre of quantum sl_n at a root of unity

It is proved that the centre Z of the simply connected quantised universal enveloping algebra over C, U_{\e,P}(sl_n), \e a primitive l-th root of unity, l an odd integer >1, has a rational field of fractions. Furthermore it is proved that if l is a power of an odd prime, Z is a unique factorisation domain.

单位根下3-循环量子Weyl代数上的单模

Simple modules over 3-cyclic quantum Weyl Algebra at roots of unity

该研究针对单位根下的3-循环量子Weyl代数，利用其作为多项式恒等式代数的性质，完成了所有单模的系统分类并算出代数中心，明确了单模维数的PI度上界。

单位根处量子矩阵代数上的单模

On Simple Modules over the Quantum Matrix algebra at roots of unity

本文在任意特征的代数闭域上研究单位根处双参数量子矩阵代数，确定其中心、计算PI度，并完成单模的同构分类。

Regular representations of the quantum groups at roots of unity

We study the bimodule structure of the quantum function algebra at roots of 1 and prove that it admits an increasing filtration with factors isomorphic to the tensor products of the dual of Weyl modules $V_λ^* \otimes V_{- ω_0 λ}^*$. As an application we compute the 0-th Hochschild cohomology of the function algebra at roots of 1.

On invariants of graphs related to quantum $\mathfrak{sl}(2)$ at roots of unity

We show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.

Quantum affine algebras at small root of unity

We study the Frobenius-Lusztig kernel for quantum affine algebras at root of unity of small orders that are usually excluded in literature. These cases are somewhat degenerate and we find that the kernel is in fact mostly related to different affine Lie algebras, some even of larger rank, that exceptionally sit inside the quantum affine algebra. This continues the authors study for quantum groups associated to finite-dimensional Lie algebras in [Len14c].

$SL_q(2)$在任意阶根单位根下的量子子群

Quantum subgroups of $SL_q(2)$ at roots of unity of arbitrary order

本文研究了$SL_q(2)$在任意阶根单位根下的量子子群分类问题，通过分析量子函数代数商来完成分类任务。

Topological invariants from quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ at roots of unity

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd order. These constructions use the notion of modified trace and relative $\mathit{G}$-modular category.

Steinberg skein identities at roots of unity

We obtain a family of skein identities in the Kauffman bracket skein module which relate Frobenius elements to Jones-Wenzl projectors at roots of unity. We view these skein identities as certain incarnations of Steinberg tensor product formulae from the theory of tilting modules of the quantum group $U_q(sl_2)$. We show that the simplest skein identities yield a short new proof of the existence of the Chebyshev-Frobenius homomorphism of Bonahon-Wong.

单位根处的量子Harish-Chandra双模与仿射Hecke范畴

Quantum Harish-Chandra bimodules at roots of unity and affine Hecke category

本文研究量子群在q为奇阶单位根时的Harish-Chandra双模范畴，将其与仿射Soergel双模范畴及非交换Springer解联系起来。

Atypical Representations of $U_{q}(sl(N))$ at Roots of Unity

We show how to adapt the Gelfand-Zetlin basis for describing the atypical representation of ${\cal U}_{\displaystyle{q}}(sl(N))$ when $q$ is root of unity. The explicit construction of atypical representation is presented in details for $N=3$.

Half-quantum groups at roots of unity, path algebras and representation type

We show that finite dimensional half-quantum groups at roots of unity corresponding to simple Lie algebras having symmetric Cartan matrix are of wild representation type, except for sl_2. Moreover, the underlying associative algebra is isomorphic to an admissible quotient of the path algebra of the Cayley graph of an abelian group. A quantum type Fourier transform enables to describe an explicit isomorphism.

量化旗 manifold 和量子群在根为 unity 时的非受限模块

Quantized flag manifolds and non-restricted modules over quantum groups at roots of unity

本文证明了量子群在根为 unity 时非受限模块的乘积公式，适用于特定条件的奇素数幂。

Symmetric polynomials vanishing on the diagonals shifted by roots of unity

For a pair of positive integers (k,r) with r>1 such that k+1 and r-1 are relatively prime, we describe the space of symmetric polynomials in variables x_1,...,x_n which vanish at all diagonals of codimension k of the form x_i=tq^{s_i}x_{i-1}, i=2,...,k+1, where t and q are primitive roots of unity of orders k+1 and r-1.

Quantum "az+b" group at roots of unity: unitary representations

All unitary representations of the quantum ``az+b'' group are found. It turns out that this quantum group is self dual i.e. all unitary representations are 'numbered' by elements of the same group. Moreover, the formula for all unitary representations involving the quantum exponential function is proven.

Iwahori-Hecke algebras of type A at roots of unity

In this paper, we explore the use of path idempotents for the Hecke algebra of type $A$ at roots of unity. For $q$ a primitive $\ell$-th root of unity we obain a non-unital imbedding of (a quotient of) the group algebra of $S_m$ into (a quotient of) the Hecke algebra $H_n(q)$ for certain $m$ and $n$. From this we recover certain instances of irreducibility criteria of Dipper, James, and Mathas, and we derive estimates on the decomposition numbers for the Hecke algebra at roots of unity. The bounds are easily computed, provide a good geometric picture of the pairs of diagrams $λ$, $μ$ for which…

单位根处量子 $SL_2$ 的余理想子代数

Coideal subalgebras of quantum $SL_2$ at roots of unity

本文分类了奇阶单位根处量子 $SL_2$ 的包络代数有限维商与坐标代数的右余理想子代数，并用生成元和关系完整刻画它们。

Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity

Let $\mathbb F$ denote an algebraically closed field and assume that $q\in \mathbb F$ is a primitive $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B,C$ and the relations assert that each of \begin{gather*} A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}} \qquad \end{gather*} commutes with $A,B,C$. We show that every finite-dimensional irreducible $\triangle_q$-module is o…

Langlands duality for representations and quantum groups at a root of unity

We give a representation-theoretic interpretation of the Langlands character duality of Frenkel and Hernandez, and show that the "Langlands branching multiplicities" for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.

On Burau representations at roots of unity

We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau's representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods.