Representations Of Quantum Groups And Constructions Of Some Admissible Quantum Affine Algebras

In this dissertation we study the structure and some representations of the quantum general linear Lie superalgebra and some novel admissible quantum affine algebras of type A1(1),Bn(1)and C2(1).In the first part of the dissertation,we investigate the representation theory of the quantum general linear superalgebra Uq(gl[(m|n)),together with uq(gl(m|n))at root of unity case,and the quantum enveloping algebra Uq(so2n).Firstly,for the quantum general linear Lie superalgebra,we explore the indecomposable submodule structures of quantum Grassmann superalgebra ?q(m|n)and its truncated objects ?q(m|n,r)at root of unity case(see[20]).Generalize the method of "intertwinedly-lifting"used in[26],we prove the indecomposability of ?q(m|n)(s)and ?q(m|n,r)(s)as Uq(gl(m|n))-modules by defining "energy degree".The Loewy filtration are described for all homogeneous subspace ?q(m|n)(s)and ?q(m|n,r)(s).Moreover,we extend the quantum Grassmann superalgebra by making tensor product with quantum dual Grassmann superalgebra Aq(m|n)defined in[20].Over this extended algebra,also as Uq(gl(m|n))-module,we construct super quantum de Rham complex(?q(m|n)(?)?q(m|n),d(?))and its subcomplex Cq(m|n,r,d(?))via defining the appropriate q-differentials.For the latter,we compute its quantum de Rham cohomology modules as the direct sum of some sign-trivial Uq(gl(m|n))-modules and determine their dimensions via some combinatorial identities.Thus,we see the non-vanishing of quantum super de Rham cohomology groups in the case when q is a root of unity,reveals the complexity of "modular representations" of small quantum supergroup uq(gl[(m|n))put forward by Lusztig.Secondly,we continue to study the realization of quantum differential operators of quantum groups or Hopf algebras.Based on the studies in papers[34],[20]and[72],for the case when q is generic,we realize the quantum enveloping algebra Uq(so2n)corresponding to Lie algebra of type D in terms of quantum differential operators defined over the quantum space of type D,such that the latter becomes a Uq(so2n)-module algebra structure and also we obtain the action formulas of uq(so2n)on the quantum divided structure for the quantum space of type D.This provides a fundamental work for the further studies on representations of uq(so2n)the Lusztig small quantum group.It is worth noting that we can further study the modular representation structure via quantum divided algebra and try to construct its corresponding quantum de Rham complex to make our knowledge system more completeIn the second part of the dissertation,we also study the construction and structure of some new admissible quantum(affine)algebras.In 1993,Damiani gave a PBW(Poincare-Birkhoff-Witt)basis in[15].In this paper,we assume that q is an indeterminate(not a root of unity).We put forward a class of admissible quantum affine algebras Uq(v1,v2,?)of affine type A1(1)(where vi ? {±1} and ? ?Q(q)*).For the case v1=v2=-1 and ?=1,we can refer to the work of Hu and Zhuang in[35].We mainly work out another case,that is,v1=v2=1 and ?=-1.The corresponding quantum affine algebra is denoted by(?)q(sl2).Two quantum affine algebras both are not isomorphic to the standard one.The difference in dealing with two algebras above is mainly reflected in the more complicate choice of root vectors(see Proposition 4.21).We gave a necessary condition to be ?=±1 for the admissible quantum affine algebra Uq,q-1(v1,v2,?)(sl2)equipped with the same quantum Weyl group as the standard one.Furthermore,We show the quantum Weyl group in the sense of Lusztig([45],[46]),generated by Ti(i=1,2)as the liftings of the generators si of the affine Weyl group of type Ai(1)acting as automorphism subgroup of(?)q(sl2),via Theorem 4.5,Proposition 4.15,Lemma 4.16 and Proposition 4.17.Also we define the imaginary root vectors and real root vectors and calculate their commutation rules.On the basis of these preparations,we achieve a PBW basis of(?)q(sl2).Because this kind of admissible quantum affine algebra(?)q(sl2)of type A1(1)has the different q-Serre relations(B5)and(B6),together with Theorem 4.3,we conjecture that its vertex operator representation theory and the finite-dimensional representation theory will be of different contents from the standard ones.On the other hand,through this work,one also can ask whether its nilpotent part is a Nichols algebra of affine type,etc.This will be a forthcoming research topic.We take quantum affine Weyl group of A1(1)type as the breakthrough to study the classification and construction of all quantum affine algebras who equipped with new admissible algebra structures as pointed Hopf algebra.Besides type A1(1),there exist new admissible quantum groups for type C2(1)and Bn(1)(containing the corresponding finite type as subalgebras).However,other type of quantum affine algebras we haven't found new objects.we describe all the admissible quantum affine algebras of type C2(1)and Bn(1)and confirm they all have a quantum Weyl group as subgroup of their automorphism group.As infinite-dimensional pointed Hopf algebra,we except to see that these new admissible quantum algebras can provide concrete examples for the classification of infinite-dimensional pointed Hopf algebra.The dissetation contains five chapters.In the first chapter,we recall some basic notions and notations,such as the(quantum)general linear Lie superalgebra,quantum(restricted)divided power algebras of type A,quantum exterior superalgebra,quantum affine(m|n)-superspace and also some arithmetic properties of q-binomials,etcIn the second chapter,we explore the indecomposable submodule structures of quantum Grassmann superalgebra ?q(m|n)and its truncated objects ?q(m|n,r)(see[20]).Moreover,we describe the construction of quantum super de Rham complex(?q(m|n)(?)?q(m|n),d(?))and compute the corresponding de Rham cohomology modules of its subcomplex Cq(m|n,r,d(?))In the third chapter,we mainly introduce the realization of the quantum enveloping algebra Uq(so2n)of type D as quantum differential operatorsIn the fourth chapter,we study the structure of the new admissible quantum affine algebra(?)q(sl2)and its quantum Weyl group.Furthermore,we determine the imaginary root vectors and real root vectors,then calculate their commutation rules based on which we achieve a PBW basis of(?)q(sl2)in terms of the Che valley generators.In the last chapter,we describe the stucture of all the admissible quantum affine algebras of type C2(1)and Bn(1).They have the same quantum Weyl group structures as the corresponding standard quantum(affine)algebras.