Representation Rings And Their Bi-Frobenius Algebra Structures

As an invariant of monoidal categories, the representation rings play a more and more important role in the study of monoidal categories, and have aroused great attention. In this thesis, we mainly deal with some structures and prop-erties of the representation ring of a finite dimensional Hopf algebra over an algebraically closed field. Especially, we study the Frobenius property of the rep-resentation ring in the framework of a bilinear form. As an application of this property, we investigate the Frobenius algebra and the Frobenius coalgebra struc-tures on a quotient of the representation ring, which leads to the construction of bi-Frobenius algebra in terms of the representation ring.Firstly, we introduce the notion of a quantum trace on the representation category of a finite dimensional Hopf algebra. We use this notion to answer the question raised by Cibils:when does the trivial module appear as a direct sum-mand of the tensor product of any two indecomposable modules? We then give some characterizations of finite dimensional indecomposable modules of quantum trace zero by virtue of almost split sequences and shall demonstrate the role of the objects of quantum trace zero in the structure of the representation ring.Secondly, according to the dimensions of morphism spaces, we define an associative and non-degenerate Z-bilinear form on the representation ring of a finite dimensional Hopf algebra. We investigate properties and structures of the representation ring in the framework of the bilinear form, including the study of the Frobenius properties, the study of some one-sided ideals, the description of the nilpotent radical and idempotents of the representation ring.Thirdly, we study the stable representation ring (the representation ring of the stable category) of a finite dimensional Hopf algebra. We show that the stable representation ring is isomorphic to a quotient of the representation ring modulo the ideal generated by the isomorphism classes of projective modules. Because of this isomorphism, the bilinear form on the representation ring induces a bilinear form on the stable representation ring. The induced bilinear form is not non-degenerate in general, we give some equivalent conditions for the non-degeneracy of the induced bilinear form. We check that the stable representation ring be- comes a group-like algebra and a bi-Frobenius algebra if the induced bilinear form is non-degenerate.In the case when the Hopf algebra is a finite dimensional spherical Hopf algebra, we study a quotient of the representation ring modulo all objects of quantum dimension zero. We show that the quotient ring can also be obtained as the representation ring of a factor category of the representation category. The quotient ring also admits a group-like algebra and a bi-Frobenius algebra structure if, in addition, the Hopf algebra is of finite representation type.Finally, as an example, we study the representation ring of a Radford Hopf algebra. We present the representation ring, the Grothendieck ring and the stable representation ring in terms of generators and relations. We also describe the nilpotent radical and idempotents of the representation ring explicitly. In particular, the stable representation ring admits a bi-Frobenius algebra structure, we describe this structure in terms of a polynomial algebra.