A New Proof of the Faithfulness of Brauer's Representation of Temperley-Lieb Algebras

Self-Adjunctions and Matrices

It is shown that the multiplicative monoids of Temperley-Lieb algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. This self-adjunction underlies the orthogonal group case of the Brauer representation of the Brauer centralizer algebra.

Brauer范畴$\boldsymbol{\textit{B}}(2)$的主图为$D_\boldsymbol{\textit{\textbackslash}infty}$

The Brauer category $\mathcal{B}(2)$ has principal graph $D_\infty$

本研究证明主图为$D_\infty$的子因子平面代数是气泡常数$\delta=2$的Brauer平面代数，该代数与$O(2)$表示范畴相关，在含$1/2$的交换环上开展，其主图编码幺半范畴信息。

划分代数的同调

The homology of the partition algebras

本文研究了划分代数的同调，证明其与对称群同调在随节点数增加的度数范围内同构，并在参数 $δ$ 可逆时在所有度数上同构，丰富了代数同调稳定性文献。

From Homological Algebra to Topology via Type B Zigzag Algebra and Heisenberg Algebra

We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel in arXiv:math/0006056. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action o…