An extension of [formula omitted] related to the alternating group and Galois orders

Abstract

In 2010, V. Futorny and S. Ovsienko gave a realization of U(gln) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of S1×S2×⋯×Sn, where Sj 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(gln), 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(gln). Finally, we discuss connections between the Gelfand-Kirillov Conjecture, A(gln), and the positive solution to Noether's problem for the alternating group.