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

We give the defining structure of the 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 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,s(G₂) to its associated object U_s^-1,_r^-1 (G₂), which give rise to the usual Lusztig symmetries defined not only on U_q (G₂) but also on the centralized quantum group U^c_q (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 quantum groups.) Some interesting (r, s)-identities holding in U_r,s(G₂) are derived from this discussion.