Abstract
AbstractThe 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 ϵ-orthogonal representation of an ϵ-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.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Algebra and Number Theory
Reference28 articles.
1. Atiyah, M., Schmid, W.: A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42, 1–62 (1977)
2. Barbasch, D., Ciubotaru, D., Trapa, P.E.: Dirac cohomology for graded afine Hecke algebras. Acta Math. 209(2), 197–227 (2012)
3. Z. Chen, Y. Kang, An analogue of the Kostant criterion for quadratic Lie superalgebras, preprint.
4. Chen, Z., Kang, Y.: Generalized Clifford theory for graded spaces. J. Pure Appl. Algebra. 220(2), 647–665 (2016)
5. D. Ciubotaru, Dirac cohomology for symplectic reection algebras, Selecta Math. (N.S.) 22 (2016), no. 1, 111–144.