Algebraic Structures on Grothendieck Groups of a Tower of Algebras

有限轨道情形下广义Weyl代数塔的Grothendieck环

Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case

本文在前期GWA权模张量积性质及无限轨道情形研究基础上，计算有限轨道支撑的GWA单与不可分解权模的张量积，给出GWA塔权模范畴Grothendieck环的生成元关系完整表示，并得到分裂Grothendieck环的部分结果。

Grothendieck rings of towers of twisted generalized Weyl algebras

Twisted generalized Weyl algebras (TGWAs) $A(R,σ,t)$ are defined over a base ring $R$ by parameters $σ$ and $t$, where $σ$ is an $n$-tuple of automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We show that, for fixed $R$ and $σ$, there is a natural algebra map $A(R,σ,tt')\to A(R,σ,t)\otimes_R A(R,σ,t')$. This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all $t$) of the Grothendieck groups of the categories of weight modules for $A(R,σ,t)$. We give presentations of these Grothendieck rings for $n=1,2$, when $R=\mathbb{C}[z]…

分拆代数塔上胞模的分支法则

Branching rules for cell modules on a tower of the partition algebras

本文利用图式化方法研究分拆代数塔上胞模的分支法则，并据此计算其格罗滕迪克环的结构常数。

Towers of graded superalgebras categorify the twisted Heisenberg double

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras…

Categorification and Heisenberg doubles arising from towers of algebras

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doub…

On Grothendieck's Conjecture about Principal Homogeneous Spaces for some Classical Algebraic Groups

In the present paper we investigate the question about the injectivity of the map F(R) --> F(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor F with transfers satisfying some additional properties. As an application we get the original proof of Special Unitary Case of Grothendieck's conjecture about principal homogeneous spaces and some other interesting examples.