On generalized characters (Q2708908)

Let \(H\) be a finite dimensional Hopf algebra over a field \(k\); \(D(H)\) denotes the Drinfeld double of \(H\), \(H^*\) the dual of \(H\). Consider the subalgebras of \(H^*\) \(\text{Ch}(H)\) (generated by the characters of \(H\)) and \(O(H^*)\) (the space of coinvariant elements of \(H^*\) under the adjoint coaction). When \(\text{char }k=0\) and \(H\) is semisimple, then \(\text{Ch}(H)=O(H^*)\); the equality does not hold in general but this justifies to call \(O(H^*)\) the subalgebra of ``generalized characters'' of \(H\). The algebra \(O(H^*)\) seems to be more suitable, for some purposes, than \(\text{Ch}(H)\). For instance, \(O(H^*)=(H^*)^H\simeq\text{End}_{D(H)}(H^*)^{\text{op}}\) as algebras [see \textit{Y. Sommerhäuser}, Das Drinfeld-Doppel und die Jones-Konstruction (preprint)]. Also, if the square of the antipode \(\mathcal S\) is inner then \(O(H^*)\simeq\text{Cocom}(H^*)\) (the space of cocommutative elements of \(H^*\)); \(O(H^*)=\text{Cocom}(H^*)\) when \({\mathcal S}^2=\text{id}\). The paper under review is a nice survey on what is known about \(O(H^*)\) [see also \textit{M. Cohen} and \textit{S. Zhu}, J. Pure Appl. Algebra 159, No. 2-3, 149-171 (2001; Zbl 0982.16030)]. There are also new results; for example, if \(B\bowtie H\) is a double crossproduct, it is shown that \(\text{End}_{B\bowtie H}(B)^{\text{op}}\simeq B^H\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].