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Generic graded contractions of Lie algebras

Mikhail V. Kochetov, Serhii D. Koval

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.

Generic graded contractions of Lie algebras

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 -graded contractions as lax monoidal structures on the identity endofunctor of the monoidal category of -graded vector spaces, we establish a functorial version of the Weimar-Woods conjecture on equivalence of generic graded contractions.

Paper Structure

This paper contains 9 sections, 13 theorems, 68 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cohomology and extensions of abelian groups
  4. Contractions and degenerations of Lie algebras
  5. Gradings and graded contractions
  6. Graded contractions via group cohomology
  7. Variety of graded contractions
  8. Graded contractions as lax monoidal functors
  9. Conclusion

Key Result

Proposition 5

The necessary and sufficient conditions for the map $\varepsilon\colon G\times G\to\mathbb F$ to define a $G$-graded contraction for all $G$-graded Lie algebras are for all $g,h,k\in G$.

Theorems & Definitions (47)

  • Definition 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Proposition 5
  • proof
  • Remark 6
  • Proposition 7
  • proof
  • Remark 8
  • ...and 37 more