Related papers: Automorphisms and isomorphism of quantum generaliz…
We construct a family of irreducible representations of the quantum plane and of the quantum Weyl algebra over an arbitrary field, assuming the deformation parameter is not a root of unity. We determine when two representations in this…
The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{A\Gamma L}}(1,q)$ whenever $q$ is sufficiently…
This paper investigates the isotropy groups of derivations on the Quantum Plane $\Bbbk_q[x, y]$, defined by the relation $yx = qxy$, where $q \in \Bbbk^*$, with $q^2\neq 1$. The main goal is to determine the automorphisms of the Quantum…
A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed…
We study the invariant theory of a class of quantum Weyl algebras under group actions and prove that the fixed subrings are always Gorenstein. We also verify the Tits alternative for the automorphism groups of these quantum Weyl algebras.
The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional C*-algebras are shown to be isomorphic to the categories of Fuss-Catalan diagrams.
For any Hecke symmetry $R$ there is a natural quantization $A_n(R)$ of the Weyl algebra $A_n$. The aim of this paper is to study some general ring-theoretic aspects of $A_n(R)$ and its relations to formal deformations of $A_n$. We also…
The objective of this paper is the proof of a conjecture of Kontsevich on the isomorphism between groups of polynomial symplectomorphisms and automorphisms of the corresponding Weyl algebra in characteristic zero. The proof is based on the…
All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$ are classified and explicitly quantized. The complete list of quantum Heisenberg--Weyl algebras so obtained includes new multiparameter deformations, most of them…
It is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$,…
Given a variety of universal algebras. A method is suggested for describing automorphisms of a category of free algebras of this variety. Applying this general method all automorphisms of such categories are found in two cases: 1) for the…
The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras…
We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of…
In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…
We discuss the $q$ deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the $q$-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily…
Associated to a finite graph $X$ is its quantum automorphism group $G$. The main problem is to compute the Poincar\'e series of $G$, meaning the series $f(z)=1+c_1z+c_2z^2+...$ whose coefficients are multiplicities of 1 into tensor powers…
This article undertakes an exploration of simple modules of 3-cyclic quantum Weyl algebra at roots of unity. Under the roots of unity assumption, the algebra becomes a Polynomial Identity algebra and the vector space dimension of the simple…
Let $R$ be a polynomial ring in $m$ variables over a field of characteristic zero. We classify all rank $n$ twisted generalized Weyl algebras over $R$, up to $\mathbb{Z}^n$-graded isomorphisms, in terms of higher spin 6-vertex…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…