Classification of Frobenius, two-step solvable Lie poset algebras

Toral posets and the binary spectrum property

We introduce a family of posets which generate Lie poset subalgebras of $A_{n-1}=\mathfrak{sl}(n)$ whose index can be realized topologically. In particular, if $\mathcal{P}$ is such a \textit{toral poset}, then it has a simplicial realization which is homotopic to a wedge sum of $d$ one-spheres, where $d$ is the index of the corresponding type-A Lie poset algebra $\mathfrak{g}_A(\mathcal{P})$. Moreover, when $\mathfrak{g}_A(\mathcal{P})$ is Frobenius, its spectrum is \textit{binary}; that is, consists of an equal number of 0's and 1's. We also find that all Frobenius, type-A Lie poset algebras…

On systems of commuting matrices, Frobenius Lie algebras and Gerstenhaber's Theorem

This work relates to three problems, the classification of maximal Abelian subalgebras (MASAs) of the Lie algebra of square matrices, the classification of 2-step solvable Frobenius Lie algebras and the Gerstenhaber's Theorem. Let M and N be two commuting square matrices of order n with entries in an algebraically closed field K. Then the associative commutative K-algebra, they generate, is of dimension at most n. This result was proved by Murray Gerstenhaber in 1961. The analog of this property for three commuting matrices is still an open problem, its version for a higher number of commuting…

某些弱整融合范畴的分类

Classification of certain weakly integral fusion categories

本文证明若干维数形式的弱整辫化融合范畴具有弱群论性，并结合维数84、90等结果，完成Frobenius-Perron维数小于120的弱整融合范畴按Morita等价的分类。

(Co)Homology of Poset Lie Algebras

We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also …

张量范畴中的弗罗贝尼乌斯子代数格子

Frobenius subalgebra lattices in tensor categories

本文将Watatani的中间子因子有限性定理推广到更广泛的单子范畴中，通过引入环境自对偶的概念，刻画了阿贝尔单子范畴中弗罗贝尼乌斯子代数偏序格子的子格，并在弱正性框架下证明了任何连通弗罗贝尼乌斯代数的相干子格都是有限的，从而显著扩展了Watatani定理。