Cohomology of products and coproducts of augmented algebras

辛曲面上的 Hecke 算子与 $\chi$-独立性

Hecke operators on symplectic surfaces and $χ$-independence

本文证明了关于拟射影辛曲面上单维层模空间的 BPS 上同调的 Toda χ-独立性猜想，并识别了与单维 Mukai 向量相关的 BPS Lie 代数，扩展了 Markman 的示性类生成定理。

Structure algebras, Hopf algebroids and oriented cohomology of a group

We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid with respect to the Hecke and the Weyl actions. We introduce new techniques (reconstruction and push-forward formula of a product, twisted coproduct, double quotients of bimodules) and apply them together with our main result to linear algebraic groups, to generalized Schubert calculus, to combinatorics of Coxeter groups and finite real root systems. As for groups, it implies that the natural Hopf-algebra structure on the algebraic oriented cohomology $h(G)$ of Levine-Morel of a split …

Cohomology of Restricted Lie Algebras

In this dissertation, we investigate the cohomology theory of restricted Lie algebras. The representation theory of restricted Lie algebras is reviewed including a description of the restricted universal enveloping algebra. In the case of an abelian restricted Lie algebra, we construct an augmented complex of free modules over the enveloping algebra that is exact in dimensions less than p and hence define the cohomology theory of these algebras in dimension less than p. In the non-abelian case, we explicitly construct cochain spaces for any coefficient module in dimensions less than 3, and giv…

The A-infinity Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category

For A a dg (or A-infinity) algebra and M a module over A, we study the image of the characteristic morphism $χ_M: HH^*(A, A) \to Ext_A(M, M)$ and its interaction with the higher structure on the Yoneda algebra $Ext_A(M, M)$. To this end, we introduce and study a notion of A-infinity centre for minimal A-infinity algebras, agreeing with the usual centre in the case that there is no higher structure. We show that the image of $χ_M$ lands in the A-infinity centre of $Ext_A(M, M)$. When A is augmented over k, we show (under mild connectedness assumptions) that the morphism $χ_k: HH^*(A, A) \to Ext…