Affine Kauffman Category And Its Quotient Categories

Suppose k is an integral domain.In this dissertation,we introduce AK,strict klinear monoidal category.It can be considered as the affinization of Kauffman category,which is introduced by Turaev[48].We call AK the affine Kauffman category.We will establish a relationship between AK and representations of quantum symplectic and orthogonal groups.Furthermore,we prove that every morphism space in affine Kauffman category is a free k-module with infinite rank.As an application,we prove that endomorphism algebras in AK(ω)are isomorphic to affine Birman-MurakamiWenzl algebras in[36].We define CK(ω,u),a quotient category of the affine Kauffman category.We prove that under the u-admissible condition,every morphism space in CK(ω,u)is a free k-module with maximal rank.In this case,endomorphism algebras in CK(ω,u)are isomorphic to cyclotomic Birman-Murakami-Wenzl algebras in[36].Assume k is an algebraically closed field.Suppose A is the locally unital k-algebra associated to CK((ω,u).In a subsequel[21],Rui-Song and I will prove that under the u-admissible condition,CK(ω,u)is a weakly triangular category.Therefore,the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified category[13].Let A-lfdmodΔ be the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration.Suppose sIk is either sI∞ or sle.Let V(u)be the integral highest weight sIk-module with the highest weight determined uniquely by u.We[21]will use A-lfdmodΔ to give a categorization for g-module V(u).In this case,g is either sIkk or(sIk,g)is the infinite rank limit of the quantum symmetric pair in the sense of Letzter[32]at the limit q to 1.