Related papers: Reflection Equation, Twist, and Equivariant Quanti…

We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank- n Hadamard type representations of the Temperley-Lieb algebra $TL_N(\sqrt n)$. They are…

Let A be any finite dimensional Hopf algebra over a field k. We specify the Tate and Tate-Hochschild cohomology for A and introduce cup products that make them become graded rings. We establish the relationship between these rings. In…

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a…

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules…

For a quasi-triangular Hopf algebra $\left( H,R\right) $, there is a notion of transmuted braided group $H_{R}$ of $H$ introduced by Majid. The transmuted braided group $H_{R}$ is a Hopf algebra in the braided category $_{H}\mathcal{M}$.…

Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the Yetter--Drinfel'd category ${}_H^H{\mathcal YD}$. We describe the bialgebras $(R{#}H)^{op}$ and…

Let $A$ be a finite dimensional symmetric Hopf algebra over a field $k$. We show that there are $A$-modules whose Tate cohomology is not finitely generated over the Tate cohomology ring of $A$. However, we also construct $A$-modules which…

Given a Hecke symmetry $R$, one can define a matrix bialgebra $E_R$ and a matrix Hopf algebra $H_R$, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to $R$. We show that for an even…

For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an $A$-module $V$ encodes the relations for tensoring the simple $A$-modules with $V$. We prove results about the eigenvalues and the right and left (generalized)…

In the Reflection Equation (RE) algebra associated with an involutive or Hecke symmetry $R$ the center is generated by elements ${\rm Tr}_R L^k$ (called the quantum power sums), where $L$ is the generating matrix of this algebra and ${\rm…

The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of…

Two different types of centrally extended quantum reflection algebras are introduced. Realizations in terms of the elements of the central extension of the Yang-Baxter algebra are exhibited. A coaction map is identified. For the special…

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a…

We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property,…

The ADR algebra $R_A$ of an Artin algebra $A$ is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the $\Delta$-filtrations of modules over RUSQ algebras and determine the projective covers of a certain…

Chevalley's theorem and it's converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker…

In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a…

Let $\mathsf{Rep}(H)$ be the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H$. Andruskiewitsch and Mombelli proved in 2007 that each indecomposable exact $\mathsf{Rep}(H)$-module category has form…

Certain quantization problems are equivalent to the construction of morphisms from "quantum" to "classical" props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof…

We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver $\overr{Q}$ and of the category of coherent sheaves on ${\mathbb P}^1$. Using this approach, we show that the…