PBW-deformations And Categorifications Of Quantum Groups

As quantum deformations of universal enveloping algebras, quantum groups were introduced independently by Drinfel’d and Jimbo in studying the quantum Yang-Baxter equation and two-dimensional solvable lattice models. Since the in-troduction of quantum groups, motivated by different reasons, many mathemati-cians extensively investigated them from different viewpoints. In this dissertation, we mainly focus on some problems related with PBW-deformations and categori-fications of quantum groups.Firstly, we consider the problem about the determination of PBW-deformations for the negative part Uq-(g) of quantized enveloping algebras Uq(g). An algorithm is established to determine when a given deformation Bq(g) of Uq-(g) is a PBW one, where the PBW-deformation theory of graded algebra and the BGG-resolution of Uq(g) play a fundamental role. As applications of our algo-rithm, we classify PBW-deformations of Uq-(g) for g of type A2and B2.Secondly, we investigate a class of PBW-deformations Bq(g) of the negative part U-q(g) of quantum groups Uq(g). This class of algebras can be viewed as a uniform description for some coideal subalgebras of Uq(g) introduced by Letzter and non-standard quantum deformation U’q(so(n,C)) of universal enveloping al-gebra U(so(n,C)) defined by Gavrilik and Klimyk. Using Kolb-Pellegrini’s braid group actions on53q(g), we define the root vectors of23g(g) and obtain some prop-erties of them, then construct some PBW bases of23g(g) via root vectors. As an application, Iorgov-Klimyk’s PBW bases for non-standard quantum deformation Uq(so(n,C)) of universal enveloping algebra U(so(n,C)) are recovered.Finally, we study categorifications of universal enveloping algebra U(so(7, C)) and Lie superalgebra osp(1|2). On one hand, we categorify the n-th. tensor power of the spin or vector representation of U/(so(7, C)) by using certain subcategories and projective functors of the BGG category of the general linear Lie algebra gln Indeed, we respectively categorify the underlying vector spaces for the n-th tensor power of the above fundamental representations of U/(so(7, C)), the actions of the generators of U(so(7,C)),and the defining relations of U(so(7, C)). On the other hand, using an exact functor between the BGG category and Harish-Chandra bimodule category of gl2, we categorify the bracket product of osp(1|2) on the odd part, where its symmetry is reflected by an isomorphism of two Harish-Chandra bimodules of gl2.