Group algebras and enveloping algebras with nonmatrix and semigroup identities

Remarks on Enveloping Semigroups

The local structures of enveloping semigroups of simple groups are investigated. All J-coirreducible connected stabilizer submonoids are determined. The notion of a navel of a reductive monoid is introduced. The cross-section lattice of the enveloping monoid is shown to be atomic. In type A, the generating series for the number of $G\times G$-orbits is found.

On the Hall algebra of semigroup representations over F_1

Let $\A$ be a finitely generated semigroup with 0. An $\A$-module over $\fun$ (also called an $\A$--set), is a pointed set $(M,*)$ together with an action of $\A$. We define and study the Hall algebra $\H_{\A}$ of the category $\C_{\A}$ of finite $\A$--modules. $\H_{\A}$ is shown to be the universal enveloping algebra of a Lie algebra $\n_{\A}$, called the \emph{Hall Lie algebra} of $\C_{\A}$. In the case of the $\fm$ - the free monoid on one generator $\fm$, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\fm$-modules) is isomorphic to Kreimer's Hopf alge…

Crystal Structure on the Category of Modules over Colored Planar Rook Algebra

Colored planar rook algebra is a semigroup algebra in which the basis element has a diagrammatic description. The category of finite dimensional modules over this algebra is completely reducible and suitable functors are defined on this category so that it admits a crystal structure in the sense of Kashiwara. We show that the category and functors categorify the crystal bases for the polynomial representations of quantized enveloping algebra $U_q(gl_{n+1})$.

经典型全正分裂实半单量子群的Haar测度

Haar measure for totally positive split real semisimple quantum groups of classical type

本文针对经典型分裂实半单李群，构造了积分其量子包络代数Borel部分的局部紧量子群，计算了其Haar权与模数据，并将其Drinfeld双视为全正半群函数代数的解析量子化。

Monodromy and Vinberg fusion for the principal degeneration of the space of G-bundles

We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of G-bundles Bun_G for an arbitrary reductive group G, and their relationship to the Langlands dual group of G. In the first part of the article we study the monodromy action on the nearby cycles sheaf along the principal degeneration of Bun_G. We describe the weight-monodromy filtration in terms of the combinatorics of the Langlands dual group of G and generalizations of the Picard-Lefschetz oscillators found in [Sch1]. Our proofs use certain local models f…