论文标题
弱三角类别的表示
Representations of weakly triangular categories
论文作者
论文摘要
在任意代数封闭的字段上，一类新的本地Unital和本地有限的维数代数$ a $。他们每个人都接受了上部有限的弱三角分解，这是上部有限三角分解的概括。承认上有限的cartan分解的任何本地联合代数均与某些特殊的本地统一代数$ a $相同，该代数$ a $ a $ a n允许上限有限的弱三角分解。可以确定的是，从Brundan-Stroppel的意义上讲，本地有限尺寸的$ -LFDMOD是$ a $ a $ modules $ a $ modules。此外，$ a $是半岛的，并且仅当其与某些愿望元素相关的centralizer subselgebras是半imimple的。此外，定义了某些内of功能，并在$ A $ -LFDMOD的子类别上进行某些代数的分类动作，该子类别由所有具有有限标准过滤的对象组成。 In the case $A$ is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, $A$ admits an upper finite weakly triangular decomposition.这导致了螺旋内代数的经典限制的表示形式分类，这些代数来自$ \ mathfrak {sl} _ \ infty $或$ \ hat {\ mathfrak {\ mathfrak {sl}} _ e $的所有可集成的最高权重模块。最后，我们研究$ a $的表示形式与详细信息相关的$ a $类别或循环组织的卡夫曼类别类别，包括在任意领域的半透明性$ a $的明确标准，以及$ -lfdmod在Brundan-Stropel的含义上是$ -LFDMOD在$ -LFDMOD上是$ -LFDMOD的最高权重类别，以及在Morita的意义上等于Morita，y Morita a $ a Morita a。 （退化）循环组合Hecke代数。
A new class of locally unital and locally finite dimensional algebras $A$ over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra $A$ which admits an upper finite weakly triangular decomposition. It is established that the category $A$-lfdmod of locally finite dimensional left $A$-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, $A$ is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined and give categorical actions of some Lie algebras on the subcategory of $A$-lfdmod consisting of all objects which have a finite standard filtration. In the case $A$ is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, $A$ admits an upper finite weakly triangular decomposition. This leads to categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of $\mathfrak {sl}_\infty$ or $\hat {\mathfrak{sl}}_e$. Finally, we study representations of $A$ associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of $A$ over an arbitrary field, and on $A$-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between $A$ and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras.
