Q-Witt Algebras, q-Lie Algebras, q-Holomorph Structure and Representations

For q generic or q = ε, a primitive lth root of 1, q-Witt algebras are described by means of q-divided power algebras. The q-Lie algebras are investigated and the q-PBW theorem for universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of q-Witt algebras is given. Based on it, the q-holomorph structure for q-Witt algebras is constructed, which interprets the realization in the context of representation theory.