Generic super-orbits in gl(m|n)* and their braided counterparts

TL;DR

This paper introduces braided varieties called braided orbits, explores their regularity criteria via Cayley-Hamilton identities, and specializes these concepts to super-orbits in gl(m|n)*, connecting quantum algebra with supergeometry.

Contribution

It develops a framework for braided varieties associated with Reflection Equation Algebras and establishes regularity criteria, extending to super-orbits in super Lie algebras.

Findings

Defined braided orbits via Reflection Equation Algebras

Established regularity criteria using Cayley-Hamilton identities

Specialized the framework to super-orbits in gl(m|n)*

Abstract

We introduce some braided varieties -- braided orbits -- by considering quotients of the so-called Reflection Equation Algebras associated with Hecke symmetries (i.e. special type solutions of the quantum Yang-Baxter equation). Such a braided variety is called regular if there exists a projective module on it, which is a counterpart of the cotangent bundle on a generic orbit O in gl(m)* in the framework of the Serre approach. We give a criterium of regularity of a braided orbit in terms of roots of the Cayley-Hamilton identity valid for the generating matrix of the Reflection Equation Algebra in question. By specializing our general construction we get super-orbits in gl(m|n)* and a criterium of their regularity.