On maximal diagonalizable Lie subalgebras of the first Hochschild cohomology

关于非标准格尔巴金-特斯拉林子代数的最高权模块

Highest weight modules with respect to non-standard Gelfand-Tsetlin subalgebras

本文探讨了复李代数$\mathfrak{gl}_n$针对非标准格尔巴金-特斯拉林子代数的最高权模块实现，并给出了此类子代数在实现上具有可对角化作用的充分条件。

Gerstenhaber brackets on Hochschild cohomology of twisted tensor products

We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in work of Buchweitz, Green, Madsen, and Solberg. We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as Gerstenhaber algebras, to the tensor product of the respective subalgebras of the Hochschild cohomology rings of the factors.

Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

For each nonzero $h\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description the Hochschild cohomology $\mathsf{HH}^\bullet(\mathsf{A}_h)$ over a field of arbitrary characteristic. In case $\mathbb{F}$ has positive characteristic, the center of $\mathsf{A}_h$ is nontrivial and …

The Batalin-Vilkovisky structure on the Tate-Hochschild cohomology ring of a group algebra

We determine the Batalin-Vilkovisky structure on the Tate-Hochschild cohomology of the group algebra $kG$ of a finite group $G$ in terms of the additive decomposition. In particular, we show that the Tate cohomology of $G$ is a Batalin-Vilkovisky subalgebra of the Tate-Hochschild cohomology of the group algebra $kG$, and that the Tate cochain complex of $G$ is a cyclic $A_{\infty}$-subalgebra of the Tate-Hochschild cochain complex of $kG$.

The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product

We consider the Belkale-Kumar cup product $\odot_t$ on $H^*(G/P)$ for a generalized flag variety $G/P$ with parameter $t \in \C^m$, where $m=\dim(H^2(G/P))$. For each $t\in \C^m$, we define an associated parabolic subgroup $P_K \supset P$. We show that the ring $(H^*(G/P), \odot_t)$ contains a graded subalgebra $A$ isomorphic to $H^*(P_K/P)$ with the usual cup product, where $P_K$ is a parabolic subgroup associated to the parameter $t$. Further, we prove that $(H^*(G/P_K), \odot_0)$ is the quotient of the ring $(H^*(G/P), \odot_t)$ with respect to the ideal generated by elements of positive de…

Locally trivial principal homogeneous spaces and conjugacy theorems for Lie algebras

For g a split semisimple Lie algebra over a field of characteristic 0, we link locally trivial principal homogeneous spaces over Spec(R) to the question of conjugacy of maximal abelian diagonalizable subalgebras of g(R).