Periplectic Brauer代数的结构、表示及其应用

In 2003, Moon introduced an algebra bearing resemblance to those of the Brauer algebra with parameter 0, which is called the periplectic Brauer algebra, as a tool to understand the mysterious representation theory of the periplectic Lie superalgebras. It has become apparent in recent years that many properties of the the Brauer algebras with parameter 0 have counterparts in the seting of periplectic Brauer algebras. A point that should be noted is that we don't know whether the periplectic Brauer algebra is a cellular algebra in the sense of Graham-Lehrer due to the missing of anti-involution, which is a standardly basis algebra in the sense of Du-Rui. On basis of the theory of the Brauer algebras with parameter 0, the project aims at investigating the structure, representations of periplectic Brauer algebras and its applications thoroughly. We will set up the fussion preocedure of the periplectic Brauer algebras, dertermine its (central) primitive idempotents, classification up to Morita equivalence, and its celluar structure. We will prove that the center of periplectic Brauers is trivial and the periplectic version of the Gavarini conjecture, which enalbes us to explore the structure and dimensions of simple modules of periplectic Brauer algebras. We will initate the study of the graded representation theory of periplectic Brauer algebras, i.e., we will provide the criterions of Koszul grading, graded hereditary, and graded cellularity of periplectic Brauer algebras. We will introduce the Schur algebra of periplectic Brauer algebra and establish the periplectic version of the Weyl-Schur duality. We expect that the project should lead to a better understanding of the structure, representations of peripletic Brauer algebra, and the representations of the periplectic Lie superalgebras.

本项目将借鉴Brauer代数，特别是参数为零的Brauer代数，的理论、方法与技巧研究periplectic Brauer代数的结构、表示及其应用。具体言之，本项目将建立periplectic Brauer代数的fusion过程，由此确定（中心）本原幂等元和Morita等价分类，证明存在fussion过程的periplectic Brauer代数是胞腔代数；证明periplectic Brauer代数的中心是平凡的；证明periplectic版本的Gavarini猜想，研究periplectic Brauer代数的单模维数、构造及标准模同态；研究periplectic Brauer代数的分次结构与表示理论；构建periplectic版本的Schur-Weyl对偶理论，由此用periplectic Brauer代数理解periplectic李超代数的最高权表示和有限维可积表示范畴等。

本项目借鉴经典Brauer代数及各类推广Brauer代数的理论、方法与技巧研究periplectic Brauer代数的结构、表示及其应用. 项目组全体成员分工合作在分圆Hecke代数的结构与表示、D型Temperley-Leib代数的半单性准则、幂等元生成代数的结构以及标准分层代数的双中心化子性质等方面取得系列学术成果: 利用Schur-Sergeev对偶的量子形式, 给出Regev公式的量子版本; 构建了分圆Hecke代数的Schur-Sergeev 对偶, 籍此给出其超Frobenius 公式; 引入分圆q-Schur 超代数, 证明分圆q-Schur 超代数是胞腔超代数; 完整刻画分圆Hecke代数权1块的Jacbson根, 从而表明Martin 的投射模根长度猜想的分圆版本不成立; 引入D型Chebychev多项式, 证明D型Temperley-Leib代数是半单的当且仅当其参数不是第二类Chebychev多项式和D型Chebychev多项式的根; 证明forked Temperley-Leib代数是半单的当且仅当其参数不是D型Chebychev多项式的根; 证明有限维代数是幂等元生成的当且仅当其任意不可约模的维数大于其自同态代数维数或其自同态代数是基域且其所有一维模的自扩张平凡. 特别地, 有限维可裂代数是幂等元生成的当且仅当其所有1维模的自扩张平凡. 由此证明整体维数有限的有限维可裂代数是幂等元生成的; 可裂域上的有限维遗传是幂等元生成的; 证明有限维标准分层代数的倾斜模在特定条件下满足双中心化子性质. 特别地, 带有保持单对象对偶函子的拟遗传代数的忠实倾斜模必满足双中心化子性质, 肯定地回答了Mazorchuk-Stroppel的极小倾斜模存在性问题; 证明分次对称胞腔代数的Higman 理想落在该代数的特定分次分支中, 且其维数不大于代数0 分支的维数; 刻画walled Brauer代数序列的Grothendieck环结构常数等.