Related papers: The isomorphism problem for multiparameter quantiz…

We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations…

Any deformation of a Weyl or Clifford algebra can be realized through some change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for…

In this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent Lie algebras. We prove that if the characteristic of the underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie algebra…

We introduce a q-analogue MW_q for the meromorphic Weyl algebra, and study the normalization problem and the symmetric powers sym^n(MW_q) for such algebra from a combinatorial viewpoint.

We deal with the classification problem of finite-dimensional representations of so called Askey--Wilson algebra in the case when $q$ is not a root of unity. We classify all representations satisfying certain property, which ensures…

Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders.…

The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical…

We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms…

We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big(…

The isomorphism problem means to decide if two given finite-dimensional simple algebras over the same centre are isomorphic and, if so, to construct an isomorphism between them. A solution to this problem has applications in computational…

Wavelets, known to be useful in non-linear multi-scale processes and in multi-resolution analysis, are shown to have a q-deformed algebraic structure. The translation and dilation operators of the theory associate with any scaling equation…

In this paper, we study a family of generalized Weyl algebras $\{\A\}$ and their polynomial extensions. We will show that the algebra $\A$ has a simple localization $\A_{\mathbb{S}}$ when none of $p$ and $q$ is a root of unity. As an…

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying…

This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. We focus on solvable Lie algebras of small dimensions over fields of arbitrary characteristic. We prove, over an…

Quantum multiparameter deformation of real Clifford algebras is proposed. The corresponding irreducible representations are found.

This article is devoted to rational equivalence for non-commutative polynomial algebras in a context including both the classical Gelfand-Kirillov problem and its quantum version. We introduce in this ``mixed'' context some reference…

We solve the isomorphism problem for non noetherian down-up algebras $A(\alpha,0,\gamma)$ by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two…

A theory of canonical basis for a two-parameter quantum algebra is developed in parallel with the one in one-parameter case. A geometric construction of the negative part of a two-parameter quantum algebra is given by using mixed perverse…

Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is…

Multiparameter quantum gl(N) is not a rigid structure. This paper defines an essential deformation as one that cannot be interpreted in terms of a similarity transformation, nor as a perturbation of the parameters. All the equivalence…