integral versus adjoint action on Hopf algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:28:02Z http://mathoverflow.net/feeds/question/103107 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103107/integral-versus-adjoint-action-on-hopf-algebra integral versus adjoint action on Hopf algebra unknown (google) 2012-07-25T15:25:32Z 2012-11-06T11:00:48Z <p>Suppose that $H$ is a finite dimensional Hopf algebra (with counit $\varepsilon$) and $T$ is a non zero right integral of $H^{\star}$ (the dual Hopf algebra). Let $ad_h$ be the adjoint action on $H$, that is, $ad_h(k)=h_1kS(h_2)$, where $S$ is the antipode map.</p> <p>When $T\circ ad_h=\varepsilon(h)T$, for all $h\in H$? Is it holds when $H$ is cosemisimple?</p> <p>For example, if $H=kG$ is the group algebra, then the equation holds for all $h\in H$. Indeed, in this case, the set of integrals (left or right) is generated by $T(g)=\delta_{g,1_G}$ and is immediate to check the equation.</p>