Two-Parameter Quantum Groups Of Type G₂

In the theory of quantum groups one deals with quantum deformation U_q(g) of the enveloping algebra U(g) of a semisimple Lie algebra g and quantum deformations kq[G] of the ring k[G] of regular functions on a semisimple algebraic group G In both cases the algebras U_q(g) or kq[G] all have an additional structure—the Hopf algebra structure which is noncommutative and noncocommutative in the sense of Drinfel'd, and it is essential that the quantum deformations are compatible with this additional structure.The algebras U_q(g) are often called quantized enveloping algebras or Drinfel'd-Jimbo algebras. These algebras (or minor modification thereof) were introduced independently by Drinfel'd and Jimbo around 1985. They were first used to construct solutions of the quantum Yang-Baxter equations. Since then they have found numerous and deep applications in areas ranging from theoretical physics, symplectic geometry, knot theory, and modular representations of reductive algebraic groups.Much complexity which was faced by the mathematicians when they research some problems made the necessity to introduce the two-parameter quantum groups and multiparameter quantum groups. For example, down-up algebras were introduced in [2,3] by G.Benkart and T.Rody. Down-up algebras exhibit many striking features including a Poincare'-Birkhoff-Witt type basis and a well-behaved representation theory. Their research showed that there exist some close connections between two-parameter quantum groups of type A and down-up algebras.There are explicit results on the two-parameter quantum groups corresponding to classical semisimple Lie algebras. But the two-parameter groups corresponding to exceptional simple Lie algebras have not been finished. So the research in this paper has important value.GBenkart and S.Witherspoon investigated two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n. Theyshowed that these quantum groups can be realized as Drinfel'd double structure. Furthermore, they constructed a corresponding R-matrix and a quantum Casimir element. They discussed isomorphisms among these quantum groups and connections with multiparameter quantum groups which were investigated by some other people. In [5], the same two authors determined the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n and gave a complete reducibility result.After this, the work about the two-parameter groups were always limited in type A case. Until the year 2001, Prof. Naihong Hu firstly found out the new structure of the two-parameter quantum groups corresponding to the orthogonal Lie algebras SO_2n+1 or SO₂n and the symplectic Lie algebra SP₂n(i.e. type B,C,D), which is of the Drinfel'd double structure. In [9] , N.GH further discussed the existence of Lusztig's isomorphisms among these quantum groups of types A,B,C, and D, and found an interesting environment condition upon which the Lusztig's symmetries exist. They constructed a corresponding R-matrix and a quantum Casimir operator in the cases of type B,C, D and investigated their finite-dimensional weight representation theory, which is of completely reducibility.The following two facts should be noticed by us. Firstly, these new two-parameter groups modulo some group-like elements are exactly the standard one-parameter quantum groups of Drinfel'd-Jimbo type. Secondly, the Drinfel'd double structure of the two-parameter groups of types A,B,C and D is helpful to the construction of R-matrix, quantum Casimir operator, and the construction of some new knot invariants through finding some Hopf subalgebras in the case of root of 1.In this paper, we investigate and firstly find the new structure (in the sense of Benkart-Witherspoon-Hu) of the two-parameter groups corresponding to the exceptional simple Lie algebras G₂. Furthermore, we investigate the Lusztig's symmetry of type G₂ and find out lots of interesting equalities of r, s.This paper is organized as fo...