Ringel-Hall Algebras Of Duplicated Tame Hereditary Algebras And Hopf Superquivers

In order to study relations between representations of quivers and Lie algebras and quantum groups, Ringel defined Hall algebras over finitary rings in 1989(see [Rin2]). Later, many working mathematicians have been trying to use Hall algebras of finite dimensional algebras over a finite field to realize Lie algebras and quantum groups, and many important and inspiring results have been obtained, see [Rin3, Rin4, Rin6, Gr2, PX1-PX3, X, DX]. The subalgebra of Hall algebra generated by the isoclasses of simple modules is called composition algebra, playing an important role in realizing Lie algebras and quantum groups, see [Rin3, Rin7, Gr2, X, GP, GZ, Zp2-Zp5, ZZ]. Also, Hall polynomials provide some convenience in calculating the structural coefficients of the corresponding Lie algebras and quantum groups. So it is interesting and important to investigate the existence of Hall polynomials, see [Rin2, Rin6, Gu, P, Zs1].Tilting modules of duplicated algebra of a finite dimensional hereditary algebra are in one-to-one correspondence with cluster tilting objects of the corresponding cluster category. This motivates further interest on this kind of algebras, see [ABST1, Zs5].Hopf algebra has attracted much research interest since the sixties of last century. In recent years with the development of quantum groups, and the strong relations between quantum groups and Yang-Baxter equations in statistics mechanics, Hopf algebra has been found to have many other connections and applications in physics. Since the nineties of last century, many people have started to study Hopf algebra using quivers. Cibils, Rosso, Green, Solberg, Van Oystaeyen and Zhang Pu have obtained many interesting and important results since then, see [Ci, CR1, CR2, GSo, VZ]. In this dissertation, we investigate the structure of Ringel-Hall algebras of dupli-cated tame hereditary algebras, obtain some Lie subalgebras induced by duplicated tame hereditary algebras; and we also develop a super version of the Hopf quiver the-ory, investigate Hopf superalgebras and quasi-Hopf superalgebras. This dissertation includes four parts altogether.In chapter 1 of this dissertation, we give an introduction, including important re-sults needed and recent developments related to this dissertation, and make a systemic exposition of our main results.In chapter 2, we investigate the structure of Ringel-Hall algebras and composition algebras of duplicated tame hereditary algebras, and we obtain some Lie subalgebras induced by duplicated tame hereditary algebras.The main results are as follows.Theorem 2.2.4 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. And let M be an indecomposable A-module. Then u[M]∈(?)(A) if and only if M is an exceptional A-module.Theorem 2.2.7 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. And let M be a non-simple indecomposable A-module. Then the element u[M]∈(?)(A) can be written as an iterated skew commutator of the isoclasses of simple A-modules.Theorem 2.3.10 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. Let X and Y be indecomposable A-modules. Then for any A-module M, there exists the Hall polynomial gXYM.In chapter 3,we develop a super version of the Hopf quiver theory, the nontion of Hopf superquivers is introduced. As an application, we give the classification of graded Hopf superalgebras and some structure theorems. The main results are as follows.Theorem 3.2.2 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ, p) admits a graded Hopf superalgebra structure if and only if (Q, p) is a Hopf superquiver of some group with respect to a super ramification datum.Corollary 3.2.3 Let G be a group, C its set of conjugacy classes and for each C∈C, let ZC denote the centralizer of one of its elements. Let R=(R0, R1)be a super ramification datum of G with R0=∑C∈CRC,0C. and R1=∑C∈CRC,1C,Denote the associated Hopf superquiver (Q(G,R0,R1),p) as (Q,p).Then the set of graded Hopf superalgebra structures on (kQ, p) with Q0≌G as groups is in one-to-one correspondence with the set of pairs of collections{(VC,0)C∈C, (VC,1)C∈C} in which VC,0 (resp. VC,1) is a kZC-module of dimension RC,0 (resp. RC,1) for all C∈C.Proposition 3.3.1 Let H be a pointed Hopf superalgebra and gr H its graded version induced by the coradical filtration. Then there is a unique Hopf superquiver (Q, p) and a graded Hopf superalgebra structure on the path supercoalgebra (kQ, p) such that gr H can be embedded into it as a sub Hopf superalgebra which contains kQ0⊕kQ1.In chapter 4, we will investigate quasi-Hopf superalgebras using Hopf superquivers. The main results are as follows.Theorem 4.2.1 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ,p) admits a graded dual quasi-Hopf superalgebra structure if and only if (Q, p) is a Hopf superquiver.Proposition 4.2.3 Let G be a group and (kG,Φ, S,α,β) a dual quasi-Hopf superalgebra. Let (Q,p)= (Q(G,R0,R1),p) be the Hopf superquiver associated to a ramification datum R=R0⊕R1 of G. Then the path supercoalgebra (kQ,p) admits a graded dual quasi-Hopf superalgebra structure with kQ0≌(kG,Φ, S,α,β) as a sub dual quasi-Hopf superalgebra if and only if kQ1 admits a kG-quasi-Hopf superbimodule structure. Moreover, the set of such graded dual quasi-Hopf superalgebra structures on the path supercoalgebra (kQ, p) is in one-to-one correspondence with the set of kG-quasi-Hopf superbimodule structures on kQ1.Proposition 4.2.4 Let H be a pointed dual quasi-Hopf superalgebra and gr H its graded version induced by the coradical filtration. Then there is a unique Hopf superquiver (Q, p) and a graded dual quasi-Hopf superalgebra structure on the path supercoalgebra (kQ, p) such that gr H can be embedded into it as a sub dual quasi-Hopf superalgebra which contains kQ0⊕kQi. Corollary 4.2.5 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ,p) admits a dual quasi-Hopf superalgebra structure (not necessarily graded) if and only if (Q, p) is a Hopf superquiver.