Primitive identities of skew derivations

The commutator δd−dδδd−dδ of two derivations δ,dδ,d always yields a derivation and must be inner if one of δ,dδ,d is inner. We generalize this to primitive expressions of skew derivations (Theorem 3.2 and Theorem 3.3). Our main results are Theorem 2.6 and Theorem 2.7. As applications, we first apply these to quantum Lie operations (Theorem 3.6). Then we extend Koryukin's counterexample [14] by adding primitive identities involving inner skew derivations (Theorem 3.7). We work in the setting of Hopf algebras and symmetric smash products. Theorem 1.3 is very useful in analyzing identities in this context.