Graded contractions of bilinear invariant forms of Lie algebras

TL;DR

This paper introduces a new method for constructing bilinear invariant forms on Lie algebras using graded contractions, applicable to various algebra types and including examples like contractions of the Killing form and toroidal contractions.

Contribution

It presents a novel approach to generating invariant forms via graded contractions, extending applicability to diverse Lie algebras and superalgebras with specific gradings.

Findings

Found $Z_2$, $Z_3$, and $Z_2\otimes\bZ_2$-contractions.

Applied contractions to the Killing form and $su(3)$.

Discussed implications for WZW actions.

Abstract

We introduce a new construction of bilinear invariant forms on Lie algebras, based on the method of graded contractions. The general method is described and the $\Bbb Z_2$-, $\Bbb Z_3$-, and $\Bbb Z_2\otimes\Bbb Z_2$-contractions are found. The results can be applied to all Lie algebras and superalgebras (finite or infinite dimensional) which admit the chosen gradings. We consider some examples: contractions of the Killing form, toroidal contractions of $su(3)$, and we briefly discuss the limit to new WZW actions.