Two-parameter Quantum Group of Exceptional Type G_2 and Lusztig's Symmetries

Abstract

We give the defining structure of two-parameter quantum group of type G_2 defined over a field {\Bbb Q}(r,s) (with r\ne s), and prove the Drinfel'd double structure as its upper and lower triangular parts, extending an earlier result of [BW1] in type A and a recent result of [BGH1] in types B, C, D. We further discuss the Lusztig's Q-isomorphisms from U_{r,s}(G_2) to its associated object U_{s^{-1},r^{-1}}(G_2), which give rise to the usual Lusztig's symmetries defined not only on U_q(G_2) but also on the centralized quantum group U_q^c(G_2) only when r=s^{-1}=q. (This also reflects the distinguishing difference between our newly defined two-parameter object and the standard Drinfel'd-Jimbo quantum groups). Some interesting (r,s)-identities holding in U_{r,s}(G_2) are derived from this discussion.