Hochschild Homology and Cohomology of Klein Surfaces

Hochschild Homology and Cohomology of Klein Surfaces

Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and ...

Hochschild Cohomology of Cubic Surfaces

We consider the polynomial algebra C[z] := C[z1, z2, z3] and the polynomial f := z 3 1 + z 3 2 + z 3 3 + 3qz1z2z3, where q ∈ C. Our aim is to compute the Hochschild homology and cohomology of the cubic surface Xf := {z ∈ C 3 / f(z) = 0}. For explicit computations, we shall make use of a method suggested by M. Kontsevich. Then, we shall develop it in order to determine the Hochschild homology an...

Second Hochschild Cohomology of Convolution Algebras

Higher order Hochschild cohomology

Following ideas of Pirashvili, we define higher order Hochschild cohomology over spheres S defined for any commutative algebra A and module M . When M = A, we prove that this cohomology is equipped with graded commutative algebra and degree d Lie algebra structures as well as with Adams operations. All operations are compatible in a suitable sense. These structures are related to Brane topology...

Symplectic Homology as Hochschild Homology

In the wake of Donaldson’s pioneering work [6], Picard-Lefschetz theory has been extended from its original context in algebraic geometry to (a very large class of) symplectic manifolds. Informally speaking, one can view the theory as analogous to Kirby calculus: one of its basic insights is that one can give a (non-unique) presentation of a symplectic manifold, in terms of a symplectic hypersu...