On the Hochschild-Kostant-Rosenberg map for graded manifolds

微分Hochschild上同调的全球同伦

Global Homotopies for Differential Hochschild Cohomologies

本文通过结合微分几何中的符号算术和余代数版本的van Est定理，构建了微分Hochschild链的显式全球同伦，将经典Hochschild-Kostant-Rosenberg映射升级为变形反演，恢复了经典定理并计算了不可及的Hochschild上同调。

Twisted Hochschild Homology of Quantum Flag Manifolds and Kähler Forms

We study the twisted Hochschild homology of quantum flag manifolds, the twist being the modular automorphism of the Haar state. We prove that every quantum flag manifold admits a non-trivial class in degree two, with an explicit representative defined in terms of a certain projection. The corresponding classical two-form, via the Hochschild-Kostant-Rosenberg theorem, is identified with a Kähler form on the flag manifold.

A Chen model for mapping spaces and the surface product

We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenb…

Differential forms on smooth operadic algebras

The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace al…