Dirac Operators and Lie Algebra Cohomology
نویسنده
چکیده
Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves [V2], [HP1], [K4]. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in many cases, but in general it has better algebraic behavior and it is more accessible for calculation.
منابع مشابه
Cohomology of aff(1|1) acting on the space of bilinear differential operators on the superspace IR1|1
We consider the aff(1)-module structure on the spaces of bilinear differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the Lie superalgebra aff(1) with coefficients in space Dλ,ν;µ of bilinear differential operators acting on weighted densities. We study also the super analogue of this problem getting the same results.
متن کاملDirac Cohomology, K-characters and Branching Laws
Inspired by work of Enright andWillenbring [EW], we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willen...
متن کاملDirac Cohomology, Unitary Representations and a Proof of a Conjecture of Vogan
The main result in this paper is a proof of Vogan’s conjecture on Dirac cohomology. In the fall of 1997, David Vogan gave a series of talks on the Dirac operator and unitary representations at the MIT Lie groups seminar. In these talks he explained a conjecture which can be stated as follows. Let G be a connected semisimple Lie group with finite center. Let K be the maximal compact subgroup of ...
متن کاملDirac Cohomology for the Cubic Dirac Operator
Let g be a complex semisimple Lie algebra and let r ⊂ g be any reductive Lie subalgebra such that B|r is nonsingular where B is the Killing form of g. Let Z(r) and Z(g) be, respectively, the centers of the enveloping algebras of r and g. Using a Harish-Chandra isomorphism one has a homomorphism η : Z(g) → Z(r) which, by a well-known result of H. Cartan, yields the the relative Lie algebra cohom...
متن کاملRepresentations up to homotopy of Lie algebroids
We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman’s BRST model ...
متن کامل