Related papers: Dirac operators and Lie algebra cohomology

Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid $\dev E\oplus \jet E$ is necessarily a Lie algebroid together with a representation on $E$. We study the geometry…

By a result of Nagy, the C*-algebra of continuous functions on the q-deformation G_q of a simply connected semisimple compact Lie group G is KK-equivalent to C(G). We show that under this equivalence the K-homology class of the Dirac…

Building upon the work of Pavel in [P. Kolesnikov, Journal of Mathematical Physics, 56, 7 (2015)], we first present the cohomology of averaging operators on the Lie conformal algebras and use it to develop the cohomology of averaging Lie…

Let G be a compact connected semisimple Lie group and let H\subset G be a closed connected subgroup such that rank(G)=rank(H) and G/H is a symmetric space. Given an irreducible representation of H, we define a Dirac operator D and determine…

The idea of using Dirac cohomology to study branching laws was initiated by Huang, Pandzi\'c and Zhu in 2013 [HPZ]. One of their results says that the Dirac cohomology of $\pi$ completely determines $\pi|_{K}$, where $\pi$ is any…

We define a pair of symplectic Dirac operators $(D^+,D^-)$ in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of $\mathbb Z/2$-graded quadratic…

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: ${\rm EI}=E_{6(6)}, {\rm EIV}=E_{6(-26)}, {\rm FI}=F_{4(4)}, {\rm…

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We…

Unifying various generalizations of the important notions of Reynolds operators, the relative cocycle weighted Reynolds operators are studied. Here cocycle weighted means the weight of the operators is given by a 2-cocycle rather than by a…

A vertex-algebraic analogue of the Lie algebroid complex is constructed, which generalizes the "small" chiral de Rham complex on smooth manifolds. The notion of VSA-inductive sheaves is also introduced. This notion generalizes that of…

In this paper, we review the construction of the Dirac operator for graded affine Hecke algebras and calculate the Dirac cohomology of irreducible unitary modules for the graded Hecke algebra of gl(n).

As an algebraic study of differential equations, differential algebras have been studied for a century and and become an important area of mathematics. In recent years the area has been expended to the noncommutative associative and Lie…

The aim of this paper is to put some recent results of Huang-Pandzic (conjectured by Vogan) and Kostant on Dirac cohomology in a broader perspective.This is achieved by introducing an induction functor in the noncommutative equivariant…

This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology…

We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying…

In this paper, we define two generalisations of Dirac operators for Drinfeld's Hecke algebra. One generalisation, Parthasarathy operators inherit the notion of the Dirac inequality. The second generalisation, warped Dirac operators are such…

This paper computes the Dirac cohomology $H_D(\pi)$ of irreducible unitary Harish-Chandra modules $\pi$ of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are…

After a brief historical survey that emphasizes the role of the algebra obeyed by the Dirac operator, we examine an algebraic Dirac operator associated with Lie algebras and Lie algebra cosets. For symmetric cosets, its ``massless''…

In this paper, we present a unified framework for studying cohomology theories of various operators in the context of pseudoalgebras. The central tool in our approach is the notion of a quasi-twilled Lie pseudoalgebra. We introduce two…

The coadjoint orbits of compact Lie groups carry many K\"ahler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi-Civita connection of the Riemannian metric, and we use…