Construction of bi-Frobenius algebras via the Benson-Carlson quotient rings

Abstract

Let $H$ be a finite dimensional spherical Hopf algebra, $r(H)$ the Green ring of $H$ and $\mathcal~{P}$ the ideal of $r(H)$ generated by all $H$-modules of quantum dimension zero. Using dimensions of negligible morphism spaces, we define a bilinear form on the Green ring $r(H)$. This form is associative, symmetric and its radical is annihilator of a certain central element of $r(H)$. After that we consider the Benson-Carlson quotient ring $r(H)/\mathcal~{P}$ of $r(H)$. This quotient ring can be thought of as the Green ring of a factor category of $H$-module category. Moreover, if $H$ is of finite representation type, the Benson-Carlson quotient ring admits group-like algebra as well as bi-Frobenius algebra structure.