Contractions and deformations of quasi-classical Lie algebras preserving a non-degenerate quadratic Casimir operator

TL;DR

This paper explores how contractions and deformations of quasi-classical Lie algebras can produce new algebra classes that preserve key quadratic Casimir operators, with implications for solutions to the Yang-Baxter equations.

Contribution

It introduces methods to generate new indecomposable quasi-classical Lie algebras via contractions and deformations that preserve non-degenerate quadratic Casimir operators.

Findings

New classes of indecomposable quasi-classical Lie algebras obtained

Conditions for preserving quadratic Casimir operators established

Connections to solutions of the Yang-Baxter equations demonstrated

Abstract

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from non-compact real simple algebras with non-simple complexification, where we impose that a non-degenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem, and obtain sufficient conditions on integrable cocycles of quasi-classical Lie algebras in order to preserve non-degenerate quadratic Casimir operators by the associated linear deformations.