Some Studies On Representations Of Rota-baxter Algebras And Quantum Affine Algebras

In mathematics research,the study of representations of any given algebraic structure plays a very important role in both its theoretical study and it application-s.This thesis focuses on the representations of Rota-Baxter algebras and minimal affinizations of type G2.It consists of five chapters.In Chapter one,we first introduces the background and motivations of this thesis.Then the main results of this thesis are summarized.Finally,some basic notations,definitions and terminologies that will be used in this thesis are presented.In Chapter two,we first introduce the concepts of free,projective,injective and flat Rota-Baxter modules.Then we give the construction of free Rota-Baxter modules and show that there are enough projective,injective and flat Rota-Baxter modules to provide the corresponding resolutions for derived functors.With emphasis on the role played by the Rota-Baxter operators,we obtain the fact that a Rota-Baxter algebra is not a free Rota-Baxter module over itself,but satisfies a universal property in a restricted sense.Chapter three focuses on the representations of the Laurent series Rota-Baxter algebra(R,P).First,we introduce the concept of "regular-singular decomposition"of an R-vector space and prove that on any R-vector space V,an(R,P)-module structure is equivalent to an regular-singular decomposition of V.Secondly,by using regular-singular decompositions of R-vector spaces,we prove that the category of representations of finite dimensional(R,P)-modules is a semisimple abelian category with exactly three isomorphism classes of irreducible objects.Furthermore,for an n-dimensional R-vector space V,we obtain that the number of GLR(V)-orbits on the set of all regular-singular decompositions of V is(n+1)(n+2)/2.We also use the result to compute the generalized class number i.e.,the number of the GLn(R)-isomorphism classes of finitely generated k[[t]]-submodules of Rn.Chapter four focuses on the representations of polynomial Rota-Baxter algebra(k[x],P).First,we show that studying the modules over the polynomial Rota-Baxter algebra(k[x],P)is equivalent to studying the modules over the Jordan plane.Thus the known results for modules of Jordan plane can immediately give their correspond-ing descriptions of(k[x],P)-modules.So we obtain the description of irreducible(k[x],P)-modules by interpreting the descriptions of the irreducible module of Jordan plane.Second,by determining the(k[x],P)-module structures on a given k[x]-module,we give the results of the direct decomposability of(k[x],P)-modules.This result al-lows us to reduce the problem of finding the structure of(k[x],P)-module(M,p)to the problem of finding the structure of its submodule.Finally,when k is an alge-braically closed field of characteristic zero,we completely and explicitly characterize the Rota-Baxter module structures on the indecomposable k[x]-modules.Chapter five focuses on minimal affinizations of type G2.We first introduce a system of equations satisfied by the q-characters of minimal affinizations of type G2 which we call the M-system of type G2.Then we prove that the M-system of type G2 contains all minimal affinizations of type G2 and only contains minimal affinizations.The equations in the M-system of type G2 are three-term recurrence relations.Thus,the M-system of type G2 is much simpler than the extended T-system of type G2 obtained by Mukhin and J.R.Li.Finally,we also interpret the three-term recurrence relations in the M-system of type G2 as exchange relations in a cluster algebra constructed by Hernandez and Leclerc.