(Deformed) Preprojective Algebras And Related Topics

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.