Derivation Algebras and 2-Cocycles of the Algebras of q-Differential Operators

Abstract

In the paper, the deriviation algebras of the associative algebras of the one-variable (resp. multivariable) q-differential operators and of their corresponding Lie algebras are determined. The completeness of the derivation algebras of the algebras of q-differential operators is also discussed. Finally, we calculate H ²(𝒟 _q (n)^?, C) for n ≥ 1, as well as H ²(g l _n (𝒟 _q ), C) under the assumption that q is transcendental over the rational numbers field Q.