The two-parameter quantum group of exceptional type G(2) and Lusztig's symmetries

摘要

We give the defining structure of the two-parameter quantum group of type G(2) defined over a field Q(r, s) (with r not equal s), and prove the Drinfel'd double structure as its upper and lower triangular parts, extending a result of Benkart and Witherspoon in type A and a recent result of Bergeron, Gao, and Hu in types B, C, D. We further discuss Lusztig's Q-isomorphisms from U-r,U-s(G(2)) to its associated object U-s-1,U-r- 1 (G(2)), which give rise to the usual Lusztig 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,U-s(G(2)) are derived from this discussion.