Notes on gamma invariants of finite dimensional Hopf algebras

Abstract

Let $H$ be a finite-dimensional, non-semisimple Hopf algebra over an algebraically closed field $\mathbf{k}$. This paper investigates the asymptotic behavior of the core of left $H$-modules through the lens of the gamma invariant $\gamma_{\mathfrak{X}}$ relative to a representation ideal $I_{\mathfrak{X}}$. We establish an equivalent characterization for the quotient of the Green ring $R_{\mathfrak{X}}$ to be a transitive fusion ring, demonstrating that transitivity is synonymous with the non-degeneracy of a naturally induced bilinear form and the collapse of the ideals $P_{+}$, $P_{-}$ and $I_{\operatorname{max}}$ into a single ideal. Furthermore, we prove that the Green ring exhibits the structure of a representation ring in the sense of Benson, provided that the square of the antipode is an inner automorphism and the equality $I_{\operatorname{max}}=I_{\operatorname{proj}}$ holds. As an explicit application of these frameworks, we analyze the Drinfeld double $D(H_4)$ of the Sweedler algebra, identifying an infinite family of distinct representation ideals and proving that the maximal gamma invariant $\gamma_{\operatorname{max}}$ induces a genuine ring homomorphism. Finally, for Hopf algebras of finite representation type under the assumption $P_{+} = P_{-} = I_{\operatorname{max}}$, we show that $\gamma_{\operatorname{max}}$ coincides precisely with the Frobenius--Perron dimension, and we explicitly compute the gamma invariants for the standard basis elements of the Green ring of the Taft algebra $H_n(q)$.