Maps between standard and principal flag orders

TL;DR

This paper explores the relationships between standard and principal flag orders and Galois orders, providing techniques for connecting their structures and morphisms, thereby simplifying their study in representation theory.

Contribution

It introduces methods to relate standard flag and Galois orders, including conditions for morphisms, properties related to differential operators, and tensor product constructions.

Findings

Established a sufficient condition for morphisms between flag and Galois orders.

Identified a property of flag orders connected to differential operators on affine varieties.

Developed tensor product constructions for flag and Galois orders.

Abstract

Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class of associative algebras that contain many important examples, such as the enveloping algebra of $\mathfrak{gl}_n$ (as well as its quantum deformation), generalized Weyl algebras, and shifted Yangians. The main motivation for introducing Galois orders is they provide a setting for studying certain infinite dimensional irreducible representations, called Gelfand-Tsetlin modules. Principal Galois orders, defined by J. Hartwig in 2017, are Galois orders with an extra property, which makes them easier to study. All of the mentioned examples are principal Galois orders. In 2019, B. Webster defined principal flag orders which in most situations are Morita equivalent to principal Galois orders, and further simplifies their study. This paper describes some techniques to connect the study pairs of standard flag and Galois orders with related data. Such techniques include: a sufficient condition for morphisms between standard flag and Galois orders to exist, a property of flag orders related to differential operators on affine varieties, and constructing tensor products of standard and principal flag and Galois orders.