一些有限性2-范畴的结构及表示理论

发布时间：2018-02-08 21:59

本文关键词： 有限性2-范畴 Duflo对合箭图单可迁2-表示 Drinfeld中心　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文刻画了有限性2-范畴的抽象Duflo对合(由每个给定的左胞腔唯一确定).同时研究了一些与(代数闭域上)有限维代数相关的有限性2-范畴的结构和表示.首先.在fiat2-范畴中.通过左胞腔对应的Duflo对合可以定义阿贝尔胞腔2-表示,这是2-版本意义上的一类不可约表示.在有限性2-范畴中,通过其主2-表示的子商形式可以定义加性胞腔2-表示.而且.在fiat情形下,对于任意的左胞腔,如此得到的加性胞腔2-表示的阿贝尔化与通过Duflo对合得到的阿贝尔胞腔2-表示是互等价的.受到flat2-范畴中的Dufl。对合定义的启发.我们在任意的有限性2-范畴中给出了类似的定义,并且证明了对于文中三类有限性2-范畴的任意左胞腔如此定义的Dufl。对合都是存在的,其中两类是与有限树箭图的路代数(简称树路代数)相关的：一类是由对偶投射函子确定的有限性2-范畴.另-类是由对偶投射函子和投射双模共同确定的有限性2-范畴.显然,后者包含前者作为其2-子范畴.不同十flat情形.有限性情形下的Duflo对合可能不落于所给左胞腔中.同时,我们描述了这两类有限性2-范畴的主2-表示的底代数的箭图,它们提供了相应的阿贝尔主2-表示作用在对象上所得范畴的一些信息,即等价于相应底代数的模范畴.其次,在有限性2-范畴中,单可迁2-表示可以看成“单”的2-表示.事实上对于任何有限性2-表示,都可构造它的一个弱的Jordan-Holder列且其弱的合成子商都是单可迁2-表示,得到相应的弱的Jordan-Holder定理.因此.具体的有限性2-范畴的单可迁2-表示的分类问题是非常有意义的.本文中,我们分类了上述与树路代数相关的第一类有限性2-范畴上的所有单可迁2-表示.同时,我们也考虑了上述三类中由有限维代数的投射双模确定的那类有限性2-范畴,我们所研究的是涉及的有限维代数非内射的情形.但是我们目前无法给出一般的分类情况.然而在其中两种较小情形下.我们分类了此类有限性2-范畴的所有单可迁2-表示.对于树路代数,我们定义了其上的可补理想,并构造了一类新的有限性2-范畴,而且分类了A。型定向箭图情形时的所有单可迁2-表示.对于这几类有限性2-范畴,我们都有结论：每个单可迁2-表示都等价于一个胞腔2-表示.然而,对于复数域上截头多项式代数的一类fiat2-范畴,此结论并不成立,它含有非胞腔2-表示的单可迁2-表示.最后,我们考虑了如何计算具体的有限性2-范畴的Drinfeld中心,它可以看成2-范畴中恒等2-函子的自同态范畴,是一个辫子monoidal范畴.在文中最后一部分,我们分别计算了上述树路代数的对偶投射函子的有限性2-范畴,An型定向箭图路代数可补理想的有限性2-范畴和截头多项式代数的fiat2-范畴的Drinfeld中心,其中一类的Drinfeld中心双等价于它的态射范畴,另一类的Drinfeld中心的不可分解对象是恒等1-态射确定的一些对.
[Abstract]:In this paper, we characterize the abstract Duflo involution of finiteness 2-category (determined by the uniqueness of each given left cell). At the same time, we study the structure and representation of some finiteness 2-categories related to finite dimensional algebras (over algebraic closed fields). In the fiat 2-category, the Abelian cell 2-representation can be defined by the Duflo involution corresponding to the left cell. This is a class of irreducible representations in the sense of 2-version. In the finiteness 2-category, the additive cell 2-representation can be defined by the subquotient form of its principal 2-representation. Moreover, in the case of fiat, for any left cell, The Abelization of the additive 2-representation and the Abelian 2-representation obtained by Duflo involution are mutually equivalent. It is inspired by the definition of Dufl-involution in the flat2-category. Have come up with a similar definition, And it is proved that the Dufl. involution of any left cell of the three finiteness 2-categories in this paper exists. Two of them are related to the path algebra of finite tree quiver (tree path algebra for short): one is finiteness 2-category determined by dual projective functor, and the other is finite determined by dual projective functor and projective bimodules. Sex 2-Category. Obviously, The latter includes the former as its 2-subcategory. Different ten flat cases. The Duflo involution in finiteness case may not fall into the given left cell. At the same time, we describe the quiver of the base algebra of the principal 2-representation of the two finiteness 2-categories. They provide some information about the category of the corresponding Abelian principal 2-representation action on the object, that is, it is equivalent to the module category of the corresponding bottom algebra. Secondly, in the finiteness 2-category, A simple transitive 2-representation can be regarded as a 2-representation of "simple". In fact, for any finite 2-representation, a weak Jordan-Holder column can be constructed and its weak compositons quotient is a simple transitive 2-representation. The corresponding weak Jordan-Holder theorem is obtained. Therefore, it is very meaningful to obtain the classification problem of the simple transitive 2-representation of the specific finiteness 2-category. In this paper, We classify all the simple transitive 2-representations of the first class of finiteness 2-category related to tree path algebras. At the same time, we also consider the class of finiteness 2-categories determined by projective bimodules of finite-dimensional algebras in the above three classes. What we are studying is the case of finite dimensional algebras that are not injective. However, we can not give a general classification at present. However, in two smaller cases, we classify all of the finiteness 2-categories. Transitive 2-representation. For tree path algebra, In this paper, we define complementary ideals on them, construct a new class of finiteness 2-categories, and classify all simple transitive 2-representations in the case of A. type directed quiver. We all have a conclusion that every simple transitive 2-representation is equivalent to a 2-representation in a cell. However, for a class of fiat2-category of truncated polynomial algebras over complex fields, this conclusion does not hold true. It contains a simple transitive 2-representation of non-cellular 2-representation. Finally, we consider how to calculate the Drinfeld center of a specific finiteness 2-category, which can be regarded as an endomorphism category of a constant iso-functor in a 2-category. Is a braided monoidal category. We calculate the finiteness of the dual projective functors of the tree path algebras mentioned above. The Drinfeld centers of the 2-category and the fiat2-category of the truncated polynomial algebras are calculated respectively. One kind of Drinfeld center is equivalent to its morphism category, the other kind of indecomposable object of Drinfeld center is some pairs of identity 1-morphism determinations.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O154.1

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