Deformation of the Heisenberg Algebra inside $gl(3,\mathbb{K})$

$q$-Heisenberg代数的完整结构分析：同调、刚性、自同构与变形

Complete Structural Analysis of $q$-Heisenberg Algebras: Homology, Rigidity, Automorphisms, and Deformations

本文研究了$q$-Heisenberg代数的结构性质，包括同调、刚性、自同构与变形，揭示其在量子变形和表示论中的重要贡献。

The Dunkl-Cherednik Deformation of a Howe duality

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{…

Supercentralizers for deformations of the Pin osp dual pair

In recent work, we examined the algebraic structure underlying a class of elements supercommuting with realization of the Lie superalgebra $\mathfrak{osp}(1|2)$ inside a generalization of the Weyl Clifford algebra. This generalization contained in particular the deformation by means of Dunkl operators, yielding a rational Cherednik algebra instead of the Weyl algebra. The aim of this work is to show that this is the full supercentralizer, give a (minimal) set of generators, and to describe the relation with the $(\mathrm{Pin}(d),\mathfrak{osp}(2m+1|2n))$ Howe dual pair.

Quantum generalized Heisenberg algebras and their representations

We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and include previously studied algebras, such as (generalized) down-up algebras. In particular, our class now includes the enveloping algebra of the $3$-dimensional Heisenberg Lie algebra and its $q$-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the cl…

正特征中受限 Lie-Rinehart 代数的形变与上同调

Deformations and Cohomology of restricted Lie-Rinehart algebras in positive characteristic

本文研究正特征下受限 Lie-Rinehart 代数的形式形变，证明 p>2 时由 Evans-Fuchs 受限上同调控制，并为 p=2 构造新的上同调复形，同时分析了受限 Heisenberg 代数的结构与上同调。

The Clifford Deformation of the Hermite Semigroup

This paper is a continuation of the paper [arXiv:0911.4725], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [arXiv:0907.3749]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.