Hopf cyclic cohomology and biderivations

Abstract:

Hopf cyclic cohomology $HC^*_{(\delta ,\sigma )}(\mathcal H)$ for a Hopf algebra $\mathcal H$ with respect to a modular pair in involution $(\delta ,\sigma )$ was introduced by Connes and Moscovici. By a biderivation $D$ on a Hopf algebra $\mathcal H$ we shall mean a linear map that satisfies the axioms for both a derivation and a coderivation on $\mathcal H$. Given a biderivation $D$ on a Hopf algebra, we define, under certain conditions, a map $L_D:HC^*_{(\delta ,\sigma )}(\mathcal H)\longrightarrow HC^*_{(\delta ,\sigma )}(\mathcal H)$. We give examples of such maps for the quantized universal enveloping algebra $\mathcal U_h(\mathfrak g)$ of a simple Lie algebra $\mathfrak g$. When $\mathcal H$ is irreducible, cocommutative and equipped with a character $\delta$ such that $(\delta ,1)$ is a modular pair in involution, we define “inner biderivations” and use these to produce a left $\mathcal H$-module structure on $HC^*_{(\delta ,1)}(\mathcal H)$. Finally, we show that every morphism $L_D:HC^*_{(\delta ,1)}(\mathcal H)\longrightarrow HC^*_{(\delta ,1)}(\mathcal H)$ induced by a biderivation $D$ on such a Hopf algebra $\mathcal H$ can be realized as a morphism induced by an inner biderivation by embedding $\mathcal H$ into a larger Hopf algebra $\mathcal H[D]$.