Hopf cyclic cohomology and biderivations

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 $\mathcal H[D]$.