Reduction of quantum systems with arbitrary first class constraints and Hecke algebras

We propose a method for reduction of quantum systems with arbitrary first class constraints. An appropriate mathematical setting for the problem is homology of associative algebras. For every such an algebra $A$ and its subalgebra B with an augmentation e there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk^{*}(A,B) which we call the Hecke algebra of the triple (A,B,e). It acts in the cohomology space H^{*}(B,V) for every left A- module V. In particular the zeroth graded component Hk^{0}(A,B) acts in the space of B- invariants of V and provides the reduction of the quantum system.Comment: 15 pages, LaTeX 2

An analogue of the operator curl for nonabelian gauge groups and scattering theory

We introduce a new perturbation for the operator curl related to connections with nonabelian gauge groups. We also prove that the perturbed operator is unitary equivalent to the operator curl if the corresponding connection is close enough to the trivial one with respect to a certain topology on the space of connections.Comment: 14 page

Strictly transversal slices to conjugacy classes in algebraic groups

We show that for every conjugacy class O in a connected semisimple algebraic group G over a field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O = codim S.Comment: 38 pages; minor modification

Semi-infinite cohomology and Hecke algebras

This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in math.QA/9805134. These new Hecke algebras are associated to triples of the form (A,B,e), where A is an associative algebra containing subalgebra B with character e. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A,B,e) naturally acts in the B-cohomology and B-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor for associative algebras and define modifications of Hecke algebras acting in semi-infinite cohomology spaces. We call these algebras semi-infinite Hecke algebras. As an example we realize the W-algebra W(g) associated to a complex semisimple Lie algebra g as a semi-infinite Hecke algebra. Using this realization we explicitly calculate the algebra W(g) avoiding the bosonization technique used by Feigin and Frenkel.Comment: 45 pages, AMSLaTeX, 1 figure using XY-pi

Conjugacy classes in Weyl groups and q-W algebras

We define noncommutative deformations $W_q^s(G)$ of algebras of functions on certain (finite coverings of) transversal slices to the set of conjugacy classes in an algebraic group $G$ which play the role of Slodowy slices in algebraic group theory. The algebras $W_q^s(G)$ called q-W algebras are labeled by (conjugacy classes of) elements $s$ of the Weyl group of $G$. The algebra $W_q^s(G)$ is a quantization of a Poisson structure defined on the corresponding transversal slice in $G$ with the help of Poisson reduction of a Poisson bracket associated to a Poisson-Lie group $G^*$ dual to a quasitriangular Poisson-Lie group. The algebras $W_q^s(G)$ can be regarded as quantum group counterparts of W-algebras. However, in general they are not deformations of the usual W-algebras.Comment: 48 pages; some arguments in the proof of Proposition 12.2 are clarifie