Generic graded contractions of Lie algebras

TL;DR

This paper investigates the structure and classification of generic graded contractions of Lie algebras using tools from group cohomology, algebraic geometry, and category theory, providing explicit classifications and functorial insights.

Contribution

It introduces a classification of generic graded contractions via an abelian group and relates them to graded degenerations and monoidal category structures.

Findings

Classified generic graded contractions with fixed support by an explicit abelian group

Described the variety of generic graded contractions as an affine algebraic variety

Established a functorial version of the Weimar-Woods conjecture

Abstract

We study generic graded contractions of Lie algebras from the perspectives of group cohomology, affine algebraic geometry and monoidal categories. We show that generic graded contractions with a fixed support are classified by a certain abelian group, which we explicitly describe. Analyzing the variety of generic graded contractions as an affine algebraic variety allows us to describe which generic graded contractions define graded degenerations of a given graded Lie algebra. Using the interpretation of generic $G$-graded contractions as lax monoidal structures on the identity endofunctor of the monoidal category of $G$-graded vector spaces, we establish a functorial version of the Weimar-Woods conjecture on equivalence of generic graded contractions.