An extension of U(gln) related to the alternating group and Galois orders

JOURNAL OF ALGEBRA

卷:569

Article

Jauch, Erich C.¹ 

[1] Univ Wisconsin, Dept Math, Eau Claire, WI 54701 USA

关键词: Alternating group;    Enveloping algebra;    Gelfand-Kirillov Conjecture;    Gelfand-Tsetlin modules;    Weight modules;   

DOI  :  10.1016/j.jalgebra.2020.10.017

来源: Elsevier

【 摘 要 】

In 2010, V. Futorny and S. Ovsienko gave a realization of U(gl(n)) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of S-1 x S-2 x ... x S-n, where S-j is the symmetric group on j variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted A(gl(n)), and show that it is a Galois ring. For n = 2, we show that it is a generalized Weyl algebra, and for n = 3 provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over A(gl(n)). Finally, we discuss connections between the Gelfand-Kirillov Conjecture, A(gl(n)), and the positive solution to Noether's problem for the alternating group. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free