Majid Double-Bosonization Theory:Reconstruction Of Two-Parameter Quantum Groups From Type A To Types A-D And G₂

For the one-parameter Drinfeld Jimbo type standard quantum groups of complex semisimple Lie algebras,Majid proposed the double-bosonization theory in 1999.In this paper,we extend this theory to the framework of two-parameter quantum groups,and realize the corresponding Majid double-bosonizati on construction for two-parameter quantum groups of types An,C3,D4 and G2,which verifies that the Majid conjecture is also valid in the two-parameter version.Specifically,the construction of two-parameter quantum group is implemented in two ways(rank-inductive and type-crossing)using irreducible representations of Ur,s(sln)and several concrete examples are given.Firstly,using the vector representation of Ur,s(sln),we realize the rank-inductive construction from Ur,s(sln)to Ur,s(sln+1).Secondly,we give the type-crossing construction in the cases of low rank from two-parameter quantum group of type A to types BCD and type A1 to the exceptional type G2.To do this,we first calculate the explicit expression of the R-matrix of Ur,s(sl2)and obtain the decomposition formula and hightest weight vectors of the tensor product of these irreducible representations.After selecting some appropriate irreducible representations of two-parameter quantum group of type A,we obtain specific forms of several lower order new R-matrices with two-parameters.Based on these special R-matrices,we give many specific constructions of two-parameter quantum groups,including A1 to B2,A2 to C3,A3 to D4 and A2 to G2.