Construct 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 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)/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.