The Research Of Some Problems On Quantum Algebras

In this paper,the main results are about the cluster algebras,geometric RTT real-ization of quantum groups and geometric realization of two parameter quantum groups.In chapter 2,we introduce and study the quantum deformations of the cluster su-peralgebras.In order to give the reasonable definition of quantum cluster superalgebra,we set up some basic algebraic structures,and prove the quantum version of the Lau-rent phenomenon for the supercase.At last,some examples are given to explain the phenomenon.In chapter 3,firstly,we use the convolution algebra on the n-steps flag variety of Fqd to realize the generators and the generate relations of the one parameter Schur algebra of Uvgln.Secondly,we find a subalgebra of the limit algebra of the Schur algbra,and prove the subalgebra is isomorphic to Uv(gln)+.That is,we give a BLM realization of the positive part of the quantum group of Uv(gln)with respect to RTT realization.In chapter 4,we give a new two parameters quantum group Uv,t(gln),prove that Uv,t(gln)is the deformation of one parameter quantum group Uv(I,·).At the same time,we also give an BLM description of the Schur-Weyl duality between two-parameters quantum algebras Uv,t(gln)and Hd(v,t).The connection of the Schur-Weyl duality for two different version of two parameter quantum group are found by us through the Galois descend approach.That is,the classic Schur-Weyl duality(Ur,s(gln),V(?)d,Hd(r,s))can be seen as a corollary of the Schur-Weyl duality(Uv,t(gln),V(?)d,Hd(v,t)).The Shur-Weylduality between the algebras Uv,t(glN)m,Uv,t(glN)m and Iwahori-Hecke algebra Hd(v,t)is established.In chapter 5,similarly to the method we used in chapter 4.Firstly,we give two new two parameter Schur algebras through the convolution algebra on the isotropic flag varieties.Then we find two new two parameters algebra(?)and(?)m over Q(v,t)through the limit algebras of the two new Schur algebra,and establish the Schur-Weyl duality between algebras(?),(?)m and two parameters Iwahori-Hecke algebra of type D.In chapter 6,the coalgebra structures of algebras(?)and(?)m are constructed throughthe BLM realizations.That is,find a Uv,t(gl2an+1)n coalgebra structure on(?)and aUv,t(gl2n)n coalgebra structure on(?)m.