（形变）预投射代数及其相关问题
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摘要
预投射代数是由Gelfand和Ponomarev在研究不含定向圈的有限箭图的表示理论时提出的一类重要的代数,它在代数表示理论、数学物理、微分几何、非交换代数等领域都起着十分重要的作用.近些年来对(形变)预投射代数的研究取得了丰硕的研究成果,但对(形变)预投射代数的表示还知之甚少.本学位论文主要考虑(形变)预投射代数的扭群代数的表示及其相关的一些问题.
首先,我们考虑了(形变)预投射代数的扭群代数Π_Q~λG上模的提升问题,即如何将(形变)预投射代数Π_Q~λG上的模提升为一个Π_Q~λG-模.对此我们考虑了一般的路代数的商代数Λ=kQ/I的情形.我们得到任意扭群代数ΛG上的模作为箭图Q的表示都是G-不变的.但反之不一定成立,在假设成立的前提下,我们准确地给出一个G-不变的Λ-模可以赋予多少种ΛG-模结构.特别地,应用到形变预投射代数的扭群代数,我们将Crawley-Boevey和Holland在形变预投射代数上定义的反射函子提升到对应的扭群代数上,得到了一些等价函子.
其次,我们考虑(形变)预投射代数的扭群代数的整个模范畴.对于任意双箭图Q及扭群代数(kQ)G,我们都可以定义一个双群形Γ_G,使得(kQ)G-mod等价于Γ_G的表示范畴,并且(kQ)G-mod中满足形变预投射关系的满子范畴等价于所有满足对应形变预投射关系的ΓG的表示组成的范畴.另一方面,由Demonet的工作可知,存在(kQ,G)的广义McKay箭图Q_G使得(kQ)G-mod等价于kQ_G-mod.在此基础上我们证明了满足形变预投射关系的ΓG的表示组成的范畴等价于kQ_G-mod中满足对应形变预投射关系的满子范畴.进而证明了形变预投射代数的扭群代数还是Morita等价于一个形变预投射代数.
对于给定的扭群代数(kQ)G,我们可以得到一个赋值图Γ和一个广义McKay箭图Q_G.最后,我们考虑Γ和Q_G的根系及对应的Kac-Moody代数之间的关系.利用(kQ)G-mod与kQ_G-mod之间的等价函子,我们可以得到一个Γ和Q_G的根系之间的映射h: Q_G→Γ.当群G是交换群时,我们证明了广义McKay箭图具有对偶性,即可以定义群G在箭图Q_G上的一种作用,使得(kQ_G, G)的广义McKay箭图为Q.利用这种对偶性,我们证明了h是一个满射.特别地,对于Γ的正实根,它在h下的原像的个数是可以确定的.此外,我们还将G在Q_G上的作用提升到Q_G对应的李代数g上,通过构造Γ的Cartan矩阵的实现,证明了Γ对应的Kac-Moody代数g(Γ)可以嵌入到g的固定点子代数中.
Preprojective algebras were introduced by Gelfand and Ponomarev to studythe representations of finite quivers without oriented cycles and nowadays occurin various areas in mathematics such as algebra representation theory, mathemat-ical physics, diferential geometry, noncommutative algebras and so on. In recentyears, the study of the (deformed) preprojective algebras achieved fruitful researchresults. However, little is known about the representation of (deformed) prepro-jective algebras. In this thesis, we study the representations of the skew groupalgebras of (deformed) preprojective algebras and some related problem.
Firstly, we consider how to construct the module of the skew group algebraof (deformed) preprojective algebra Π_Q~λG, i.e., how to lift a Π_Q~λG-module to a Π_Q~λG-module. We consider the general case of the quotient of the path algebra Λ=kQ/I, and obtain that any ΛG-module is a G-stable Λ-module. But the converseis not necessarily true. Assume it is true, we can give accurately the ΛG-modulestructure of a G-stable Λ-module. In particular, for the skew group algebras ofdeformed preprojective algebras, we extend the reflection functor for deformedpreprojective algebras defined by Crawley-Boevey and Holland to it’s skew groupalgebras.
Secondly, we consider the module category of the skew group algebras of (de-formed) preprojective algebras. For any double quiver Q and skew group algebra(kQ)G, we can define a double group species ΓG, such that the category (kQ)G-mod is equivalent to the representation category of ΓG, and the full subcategoryof (kQ)G-mod consists the module satisfying the deformed preprojective relationsis equivalent to the category consists all the representations of ΓGsatisfying thecorresponding deformed preprojective relations. On the other hand, by Demonet’sresult, there exists a generalized McKay quiver Q_Gsuch that (kQ)G-mod is equiv-alent to kQ_G-mod. Accordingly, we prove that the full subcategory of kQ_G-modconsists the module satisfying the deformed preprojective relations is equivalent tothe category consists the ΓG-representations satisfying the corresponding deformed preprojective relations. And then prove that the skew group algebra of (deformed)preprojective algebra is Morita equivalent to a (deformed) preprojective algebra.
For a given skew group algebra (kQ)G, there is a valued graph Γ and ageneralized McKay quiver Q_G. Finally, we consider the relationship between theroot systems and the Kac-Moody algebras corresponding to Γ and Q_G. Usingthe equivalence functor between (kQ)G-mod and kQ_G-mod, we construct a maph: Q_G→Γbetween the root systems of Γ and Q_G. If G is Abelian, we can givethe duality of the generalized McKay quiver. That is to say, we define an actionof G on Q_G, such that the generalized McKay quiver of (kQ_G, G) is Q. By thisduality, we prove that h is a surjection. In particular, for any positive real root ofΓ, the number of it’s preimage can be determined. Moreover, we lift the action ofG on Q_Gto it’s Lie algebra g, and construct a realization of the Cartan matrixof Γ, such that Kac-Moody algebra g(Γ) can be embedded into the fixed-pointsubalgebra of g.
首先,我们考虑了(形变)预投射代数的扭群代数Π_Q~λG上模的提升问题,即如何将(形变)预投射代数Π_Q~λG上的模提升为一个Π_Q~λG-模.对此我们考虑了一般的路代数的商代数Λ=kQ/I的情形.我们得到任意扭群代数ΛG上的模作为箭图Q的表示都是G-不变的.但反之不一定成立,在假设成立的前提下,我们准确地给出一个G-不变的Λ-模可以赋予多少种ΛG-模结构.特别地,应用到形变预投射代数的扭群代数,我们将Crawley-Boevey和Holland在形变预投射代数上定义的反射函子提升到对应的扭群代数上,得到了一些等价函子.
其次,我们考虑(形变)预投射代数的扭群代数的整个模范畴.对于任意双箭图Q及扭群代数(kQ)G,我们都可以定义一个双群形Γ_G,使得(kQ)G-mod等价于Γ_G的表示范畴,并且(kQ)G-mod中满足形变预投射关系的满子范畴等价于所有满足对应形变预投射关系的ΓG的表示组成的范畴.另一方面,由Demonet的工作可知,存在(kQ,G)的广义McKay箭图Q_G使得(kQ)G-mod等价于kQ_G-mod.在此基础上我们证明了满足形变预投射关系的ΓG的表示组成的范畴等价于kQ_G-mod中满足对应形变预投射关系的满子范畴.进而证明了形变预投射代数的扭群代数还是Morita等价于一个形变预投射代数.
对于给定的扭群代数(kQ)G,我们可以得到一个赋值图Γ和一个广义McKay箭图Q_G.最后,我们考虑Γ和Q_G的根系及对应的Kac-Moody代数之间的关系.利用(kQ)G-mod与kQ_G-mod之间的等价函子,我们可以得到一个Γ和Q_G的根系之间的映射h: Q_G→Γ.当群G是交换群时,我们证明了广义McKay箭图具有对偶性,即可以定义群G在箭图Q_G上的一种作用,使得(kQ_G, G)的广义McKay箭图为Q.利用这种对偶性,我们证明了h是一个满射.特别地,对于Γ的正实根,它在h下的原像的个数是可以确定的.此外,我们还将G在Q_G上的作用提升到Q_G对应的李代数g上,通过构造Γ的Cartan矩阵的实现,证明了Γ对应的Kac-Moody代数g(Γ)可以嵌入到g的固定点子代数中.
Preprojective algebras were introduced by Gelfand and Ponomarev to studythe representations of finite quivers without oriented cycles and nowadays occurin various areas in mathematics such as algebra representation theory, mathemat-ical physics, diferential geometry, noncommutative algebras and so on. In recentyears, the study of the (deformed) preprojective algebras achieved fruitful researchresults. However, little is known about the representation of (deformed) prepro-jective algebras. In this thesis, we study the representations of the skew groupalgebras of (deformed) preprojective algebras and some related problem.
Firstly, we consider how to construct the module of the skew group algebraof (deformed) preprojective algebra Π_Q~λG, i.e., how to lift a Π_Q~λG-module to a Π_Q~λG-module. We consider the general case of the quotient of the path algebra Λ=kQ/I, and obtain that any ΛG-module is a G-stable Λ-module. But the converseis not necessarily true. Assume it is true, we can give accurately the ΛG-modulestructure of a G-stable Λ-module. In particular, for the skew group algebras ofdeformed preprojective algebras, we extend the reflection functor for deformedpreprojective algebras defined by Crawley-Boevey and Holland to it’s skew groupalgebras.
Secondly, we consider the module category of the skew group algebras of (de-formed) preprojective algebras. For any double quiver Q and skew group algebra(kQ)G, we can define a double group species ΓG, such that the category (kQ)G-mod is equivalent to the representation category of ΓG, and the full subcategoryof (kQ)G-mod consists the module satisfying the deformed preprojective relationsis equivalent to the category consists all the representations of ΓGsatisfying thecorresponding deformed preprojective relations. On the other hand, by Demonet’sresult, there exists a generalized McKay quiver Q_Gsuch that (kQ)G-mod is equiv-alent to kQ_G-mod. Accordingly, we prove that the full subcategory of kQ_G-modconsists the module satisfying the deformed preprojective relations is equivalent tothe category consists the ΓG-representations satisfying the corresponding deformed preprojective relations. And then prove that the skew group algebra of (deformed)preprojective algebra is Morita equivalent to a (deformed) preprojective algebra.
For a given skew group algebra (kQ)G, there is a valued graph Γ and ageneralized McKay quiver Q_G. Finally, we consider the relationship between theroot systems and the Kac-Moody algebras corresponding to Γ and Q_G. Usingthe equivalence functor between (kQ)G-mod and kQ_G-mod, we construct a maph: Q_G→Γbetween the root systems of Γ and Q_G. If G is Abelian, we can givethe duality of the generalized McKay quiver. That is to say, we define an actionof G on Q_G, such that the generalized McKay quiver of (kQ_G, G) is Q. By thisduality, we prove that h is a surjection. In particular, for any positive real root ofΓ, the number of it’s preimage can be determined. Moreover, we lift the action ofG on Q_Gto it’s Lie algebra g, and construct a realization of the Cartan matrixof Γ, such that Kac-Moody algebra g(Γ) can be embedded into the fixed-pointsubalgebra of g.
引文
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