Automorphic Equivalence in the Varieties of Representations of Lie algebras

TL;DR

This paper investigates the relationship between geometric and automorphic equivalence in varieties of Lie algebra representations over a characteristic zero field, identifying the group that measures their differences and providing a specific example.

Contribution

It introduces a new group that quantifies the difference between geometric and automorphic equivalence in Lie algebra representations and offers a concrete example illustrating this distinction.

Findings

Calculated the group measuring differences between equivalences

Identified automorphically equivalent but not geometrically equivalent representations

Provided a specific example in a subvariety

Abstract

In this paper we consider the very wide class of varieties of representations of Lie algebras over the field k, which has characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations of these varieties. We calculate the group, which measures the difference between the geometric equivalence and automorphic equivalence of representations of theses varieties. In Section 5, we present one example of the subvariety of the variety of all the representations of the Lie algebras over the field k, and two representations from these variety which are automorphically equivalent but not geometrically equivalent.