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The two-parameter quantum group of exceptional type G2 and Lusztig’s symmetries

We give the defining structure of two-parameter quantum group of type G2 defined over a field Q(r, s) (with r 6 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 Ur,s(G2) to its associated object Us−1,r−1(G2), which give rise to the usual Lusztig's symmetries defined not only on Uq(G2) but also on the centralized quantum group U c(G2) 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 quan- tum groups). Some interesting (r, s)-identities holding in Ur,s(G2) are derived from this discussion.