Commutative quotients of finite W-algebras

Commutative algebras and representations of the category of finite sets

We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.

基于组合零点定理的交换半单群代数Wedderburn分解

Wedderburn decomposition of commutative semisimple group algebras using the Combinatorial Nullstellensatz

该研究从几何角度出发，利用Combinatorial Nullstellensatz得到有限阿贝尔群及有限域上半单交换群代数的Wedderburn分解的单分量，完善了相关代数结构的刻画。

On symmetric quotients of symmetric algebras

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring ${\mathcal O}$. Using elementary methods, we show that if an ordinary irreducible character $χ$ of a finite group $G$ gives rise to a symmetric quotient over ${\mathcal O}$ which is not a matrix algebra, then the decomposition numbers of the row labelled by $χ$ are all divisible by the characteristic $p$ of the residue field of ${\mathcal O}$.

Representations of Hecke algebras on quotients of path algebras

Let $(W,S)$ be a Coxeter system. A $W$-graph encodes a representation of the Hecke algebra $\mathcal{H}$ of $W$. We construct universal representations of multi-parameter Hecke algebras on certain quotients of path algebras, and study their relationships with $W$-graph representations. We also study the quotients of path algebras on their own, motivated by one example where the quotient path algebra is isomorphic to an ideal of Lusztig asymptotic Hecke algebra. Finally, we describe a method to obtain a generating set for the ideals by which we quotient the path algebras.

On a new class of the commutative subalgebras of the maximal Gel'fand-Kirillov dimension in the universal enveloping algebra of a simple Lie algebra

New commutative subalgebras of the maximal Gel'fand-Kirillov dimension in the universal enveloping algebras of classical Lie algebras gl(n) and so(n) are constructed. In the case of sp(n) Gel'fand-Tsetlin algebra is extended to a maximally commutative one.

Special multserial algebras are quotients of symmetric special multiserial algebras

In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.

The argument shift method and maximal commutative subalgebras of Poisson algebras

Let $S$ be the symmetric algebra of an algebraic Lie algebra. We provide a sufficient condition for the maximality of Poisson commutative subalgebras of $S$ obtained by the argument shift method.

Tilting modules and exceptional sequences for leaf quotients of type A zig-zag algebras

We classify generalized tilting modules and full exceptional sequences for the family of quasi-hereditary quotients of type A zig-zag algebras and for a related family of algebras. We also give a characterization of these quotients as quasi-hereditary algebras with simple preserving duality that are 'close' to self-injective algebras.

Finite dimensional quotients of Hecke algebras

Let W be a complex reflection group. We prove that there is the maximal finite dimensional quotient of the Hecke algebra H_q(W) of W and that the dimension of this quotient coincides with |W|. This is a weak version of a Broué-Malle-Rouquier conjecture from 1998. The proof is based on categories O for Rational Cherednik algebras.

埃尔米特交换 *-代数由无界算子的表示

Representations of Hermitian Commutative *-Algebras by Unbounded Operators

本文针对埃尔米特交换幺 *-代数，证明了由预希尔伯特空间中可能无界的算子给出的幺表示的谱定理，并指出可数生成情形下已有更优结果。

A maximal cubic quotient of the braid algebra

We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. It is finite-dimensional for at least n at most 5 and we investigate its module structure in this range. We also investigate the proper quotients of it that appear in the realm of quantum groups, and describe another maximal quotient related to the usual Hecke algebras. Finally, we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by Vogel. We…

Homotopy Invariant Commutative Algebra over fields

These notes illustrates the power of formulating ideas of commutative algebra in a homotopy invariant form. They can then be applied to derived categories of rings or ring spectra. These ideas are powerful in classical algebra, in representation theory of groups, in classical algebraic topology and elsewhere. The notes grew out of a series of lectures given during the `Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM (Barcelona) in Spring 2015.

Representations of finite-dimensional quotient algebras of the 3-string braid group

We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.