PBW-deformations Of Quantum Groups Of Rank 2 And Their Symmetries

In the past twenty years,the PBW-deformation theory of connected graded algebras(especially for the algebras that have good homological properties such as Koszul algebras,d-Koszul algebras)has attracted considerable attentions and been widely studied.For general connected graded algebras,Cassidy and Shelton established the sufficient and necessary condition,i.e.,the Jacobi condition,for determining PBW-deformations.In essential,the Jacobi condition is equivalent to the problem about solving a series of linear equations,and its complexity is related with an important homological invariant which is called the complexity of algebra.Quantum groups are noncommutative and non-cocommutative noetherian Hopf algebras,while their negative parts are noetherian connected graded algebras.They both have good homological properties.In fact,for an arbitrary semisimple Lie algebra g,quantum group Uq(g)and its negative part Uq-1(g)are both Artin-Schelter regular algebras and skew Calabi-Yau algebras.On the other hand,the coideal subalgebras of quantum groups appearing in the theory of quantum symmetric pairs afford many examples for PBW-deformations of Uq-1(g)(abbr.PBW-deformations of quantum group),in which there are some symmetry phenomena.Motivated by the above facts,in this dissertation we mainly focus on the PBW-deformations of quantum groups of rank 2 and their symmetries.Using the Jacobi condition established by Cassidy and Shelton,we explicitly characterize all the PBW-deformations of quantum groups of type A2 and B2 by giving the sufficient and necessary conditions satisfied by the structure coefficients,and then characterize the symmetric PBW-deformations.For the type G2 case,we prove that the deformations of Uq-(G2)which are obtained just by altering the degree 0 part are PBW-deformations,and then describe the sufficient and necessary condition when they are symmetric.