Principal Galois orders and Gelfand-Zeitlin modules

TL;DR

This paper introduces principal Galois orders within skew monoid rings, providing criteria for their identification, constructing Gelfand-Zeitlin modules, and applying these concepts to various algebraic structures including W-algebras and quantum OGZ algebras.

Contribution

It defines principal Galois orders, establishes criteria for Galois rings to be Galois orders, and constructs canonical simple Gelfand-Zeitlin modules for these algebras.

Findings

Principal Galois orders contain Galois rings with maximal commutative Gelfand-Zeitlin subalgebras.

Constructed canonical simple Gelfand-Zeitlin modules for principal Galois orders.

Proved that quantum OGZ algebras are principal Galois orders, confirming the Mazorchuk-Turowska conjecture.

Abstract

We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order. As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type $A$, and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative. Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if $q$ is not a root of unity, the Gelfand-Zeitlin subalgebra of $U_q(\mathfrak{gl}_n)$ is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.