Structure And Realizations Of Multiparameter Quantum Groups

In this dissertation, inspired by the recent research on two-parameter quantum groups, we investigate a family of multiparameter quantum groups U_q((?)_A), defined by C. Fronsdal [45](or see V. Kharchenko [81], M. Rosso [121]), where (?)_A is a Kac-Moody algebra with symmetrizable generalized Cartan matrix A = (a_ij)_i,j∈I and the parameter matrix q = (Q_ij)_i,j∈I satisfies q_ijq_ji = q_ii^aij, (?) i,j∈I. U_q((?)_A) contains the well-known one-parameter Drinfel'd-Jimbo quantum groups, two-parameter quantum groups appeared in [9, 11, 16, 63, 68, 69] and many other multiparameter quantum groups as special cases. We always assume that q_ii(i∈I) is not a root of unity.Firstly, we show that U_q((?)_A) has the Hopf structure with the triangular decomposition, (as vector spaces), where U_q⁺,U_q⁰,U_q^- are subalgebras of U_q((?)_A). U_q((?)_A) has the Drinfel'd double structure,(as Hopf algebras),where U_q^≤0,U_q^≥0 are Hopf subalgebras of U_q((?)_A), <,>_q is the skew-Hopf paring between U_q^≤0 and U_q^≥0. The positive part U_q⁺ of U_q((?)_A) has a U_q((?)_A) module-algebra structure. The action is defined by, for all μ∈Q,i∈I, x∈ (U_q⁺)_β,where (?) is the skew derivation of U_q((?)_A). We study representations of U_q((?)_A), which is quite similar to that of Kac-Moody algebras(one or two parameter quantum groups). The category (?)t is introducedand studied, we prove that <, >_q is non-degenerate. Moreover, we show that the irreducible highest weight U_q((?)_a)-module V^q(λ) belongs to to the category (?) if and only if λ∈ A⁺, and (?) is a semi-simple braided tensor category.Secondly, according to the deformation theory of Hopf algebras, we show that U_q((?)_A) is twist-equivalent to one-parameter Drinfel'd-Jimbo quantum group U_q((?)_A),(as Hopf algebras), where σ is a Hopf 2-cocycle of U_q((?)_A). U_q((?)_A) is twist-equivalent to one-parameter Drinfel'd-Jimbo quantum group U_q((?)_A),(as bi-grading Hopf algebras), where σ' is a 2-cocycle of the free abelian group Q × Q. As application, we give a new proof for the existence of skew Hopf pairing. Furthermore, we show that(as braided tensor categories),where (?) is the corresponding module category for one-parameter quantum group U_q((?)_A).Finally, we study the realizations of the positive part U_q⁺. We study the multiparameter version of Green-Lusztig-Rosso's theory. We show that (as algebras),where U_q is the subalgebra of multiparameter quantum shuffle algebra, f_q is the multiparameter Lusztig algebra. We study the multiparameter version of Green-Lusztig-Ringel's theory(non-generic). We show that (as algebras),where (?)_v(∧_k,R) is the composition subalgebra of multiparameter Ringel-Hall algebra(non-generic case).