For an algebra B with an action of a Hopf algebra H we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg’s theorem relating...

For the root system of type A we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application of our construction we describe a generalization of the equivariant Pieri rule for double Schubert polynomials. For a general finite Coxeter syste...

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...

Let g be a finite dimensional simple Lie algebra over the complex numbers C. Consider the exterior algebra R := ∧ (g⊕ g) on two copies of g. Then, the algebra R is bigraded under R = ∧p(g)⊗∧q(g). The diagonal adjoint action of g gives rise to a g-algebra structure on R compatible with the bigrading. There are three ‘standard’ copies of the adjoint representation g in R. Let J be the (bigraded) ...

Introduction 2 1. Preliminaries on conformal algebras and modules 4 2. Basic definitions 9 3. Extensions and deformations 13 4. Homology 17 5. Exterior multiplication, contraction, and module structure 18 6. Cohomology of conformal algebras and their annihilation Lie algebras 19 6.1. Cohomology of the basic complex 19 6.2. Cohomology of the reduced complex 21 6.3. Cohomology of conformal algebr...

Let A+(k) denote the ring l~[ t] / t k+t and let fr be a reductive complex Lie algebra with exponents m 1 . . . . . m,. This paper concerns the Lie algebra cohomology of ~| considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we call weight, is inherited from the obvious grading of f f | § (k)). We conjecture that this Lie algebra cohomology is a...

The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue Poisson structures. We construct an $$L_\infty $$ -algebra whose Maurer–Cartan elements are given operators. This leads to cohomology associated operator. can be seen Hochschild certain algebra with coefficients in sui...

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental (“topological”) information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of...

In this paper, we formulate a generalization of the classical BRST construction which applies to the case of the reduction of a poisson manifold by a submanifold. In the case of symplectic reduction, our procedure generalizes the usual classical BRST construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold, i.e. the case of reducible “first c...

It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extent this is still true. We give an explicit description of the Ext-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a t...