论文标题
量子概括的海森堡代数及其表示形式
Quantum generalized Heisenberg algebras and their representations
论文作者
论文摘要
我们介绍并研究了一类新的代数，我们将其命名\ textit {量子概括性海森贝格代数}，并用$ \ Mathcal {h} _q（f，g）$表示，与普遍的Heisenberg代数相关，但允许更多的自由参数，以及以前的应用程序，以及综合范围的范围，并包括范围的范围（综合范围）（综合范围）（综合范围）（综合范围）（综合范围）（遍布范围）（综合范围）（遍布范围）（遍布范围）（遍布范围）（遍布范围）（遍布范围）（遍布范围）（遍布范围）（遍布范围）（遍布范围），代数。特别是，我们的班级现在包括$ 3 $二维的海森伯格lie代数及其$ q $ po-$ performation的代数，它们都无法实现为广义的海森伯格代数。 本文主要侧重于量子概括的海森贝格代数的有限维不可减少表示的分类，这项研究揭示了其丰富的结构。尽管这些代数并非一般的Noetherian，但它们的表示仍然保留了一些谎言理论的风味。我们在一个任意特征的领域工作，尽管我们的结果在表示形式上要求其在代数上关闭。
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 classification of finite-dimensional irreducible representations of quantum generalized Heisenberg algebras, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain some Lie-theoretic flavor. We work over a field of arbitrary characteristic, although our results on the representations require that it be algebraically closed.
