Cubic Dirac operators and the strange Freudenthal-de Vries formula for colour Lie algebras

Abstract

The aim of this paper is to define cubic Dirac operators for colour Lie algebras. We give a necessary and sufficient condition to construct a colour Lie algebra from an $\epsilon$-orthogonal representation of an $\epsilon$-quadratic colour Lie algebra. This is used to prove a strange Freudenthal-de Vries formula for basic colour Lie algebras as well as a Parthasarathy formula for cubic Dirac operators of colour Lie algebras. We calculate the cohomology induced by this Dirac operator, analogously to the algebraic Vogan conjecture proved by Huang and Pandžić.

References in corpus (3)

• The Kostant invariant and special $\epsilon$-orthogonal representations for $\epsilon$-quadratic colour Lie algebras
• Kostant's cubic Dirac operator of Lie Superalgebras
• An analogue of the Kostant criterion for quadratic Lie superalgebras

Cited by in corpus (4)

• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
• A classification of lowest weight irreducible modules over ${\mathbb{Z}}_2^2$-graded extension of $osp(1|2)$
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras