Classification Of U_q(sl₂)-Module Algebra Structures On The Quantum Plane C_q'[x,y]

In order to further consider the more complex quantum group modular algebra structure on the quantum plane,we describe the classification Uq(sl2)-module algebra structures on Cq'[x,y],when q?q'.The layout is as follows.There are four chapters,the first of which is a short introduction of some basic concepts and results.We briefly recall some notations about Hopfmodule algebras,quantum groups Uq(sl2),and the definitions of quantum polynomial algebra,quantum plane and so on.Similar to the discussions of Duplij and Sinel'shchikov[21],we introduce a full action matrix M,an action k-matrix,an action of Mef-matrix,symbol matrix and the i-th homogeneous component symbol matrix and etc.Secondly,when q?q',we compute 0-th,1-th component of the symbol matrix i.e(Mef)0,(Mef)1.One knows that every monomial of Cq'[x,y]is an eigenvector for k,and one can calculate the associated eigenvalue.Moreover,according to the 0-th homogenous component of the full action matrix M and some relations,one can get 6 cases of(Mef)0,5 of which determine the specific values of ?,?.In a similar way,we also consider the 1-th homogenous component of M,and receive 5 cases of(Mef)1.There are 4 cases of(Mef)1 which fix the precise values of ?,?.Then we introduce symbolic matrix series.Because each such matrix determines a pair of specific weight constants ?,?,and we exclude all the empty series.Finally,we get 5 non-empty series.Finally,we turn to discuss the remaining 5 non-empty series one by one.Consequently,we obtain the classification Uq(sl2)-module algebra structures on Cq'[x,y].