Related papers: Homological methods in semi-infinite contexts
It is known that the semi-infinite cohomology spaces of the infinitely twisted nilpotent subalgebra in an affine Lie algebra $g$ with coefficients in an integrable simple module over the affine Lie algebra have a base enumerated by elements…
We discuss the category $\cal I$ of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some…
A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac-Moody groups and Lie…
The goal of the present paper is to obtain new free field realizations of affine Kac-Moody algebras motivated by geometric representation theory for generalized flag manifolds of finite-dimensional semisimple Lie groups. We provide an…
By a local geometric Langlands correspondence for a complex reductive group G we understand a construction which assigns to a local system on the punctured disc for the Langlands dual group of G, a category equipped with an action of the…
In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are…
In this paper, we give a new series of coboundary operators of Hom-Lie algebras. And prove that cohomology groups with respect to coboundary operators are isomorphic. Then, we revisit representations of Hom-Lie algebras, and generalize the…
We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld. We generalize in this way the Dirac cohomology theory for Lusztig's graded affine Hecke algebras. We…
In this paper we develop a new homology theory of associative algebras called semiinfinite cohomology in a derived category setting. We show that in the case of small quantum groups the zeroth semiinfinite cohomology of the trivial module…
Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the…
Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional''…
In this paper we use the quantization of fields based on Geometric Langlands Correspondence \cite{diep1} to realize the automorphic representations of some concrete series of groups: for the affine Heisenberg (loop) groups it is reduced to…
We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the…
We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the…
Let $\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, $\mu$ a diagram automorphism of $\mathfrak{g}$ and $L(\mathfrak{g},\mu)$ the loop algebra of $\mathfrak{g}$ associated to $\mu$. In this…
We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are…
In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations…
The article covers developments in the representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and…
We give a new interpretation and proof of the "quasi-particle" type character formulas for integrable representations of the simply-laced affine Kac-Moody algebras through a new "semi-infinite" construction of such representations. We…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…