On geometric degenerations and Gerstenhaber formal deformations

TL;DR

This paper explores how algebraic structures like associative and Lie algebras degenerate, linking these processes to Gerstenhaber deformations, and applies homological algebra to classify degenerations of 3-dimensional Lie algebras.

Contribution

It provides a new description of degeneration relations via Gerstenhaber deformations and applies homological algebra to classify orbit closures of 3-dimensional Lie algebras.

Findings

Orbit closure of 3D Lie algebras determined by homological algebra.

N-Koszul property preserved under degeneration for all N≥2.

Degeneration relations characterized in terms of formal deformations.

Abstract

We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to show how the orbit closure of the $3$-dimensional Lie algebras can be determined using homological algebra. For the case of finite dimensional associative algebras, we prove that the $N$-Koszul property is preserved under the degeneration relation for all $N\geq2$.