\(K\)-theory of braided monoidal categories (Q1811047)

In this paper the following result is proven: The algebraic K-theory of a braided monoidal abelian category with exact tensor product is a Gerstenhaber algebra. Here algebraic K-theory is considered as a graded abelian group. The reader might want to compare with the result of \textit{G. Dunn} [K-Theory 9, 591-605 (1995; Zbl 0845.19001)] that the algebraic K-theory spectrum of a braided bimonoidal category with cofibrations and weak equivalences is an \(E_2\)-ring spectrum. Let \(P_n\) be the \(n\)-th pure braid group. These assemble to an operad \(P_*\) whose algebras, when interpretation as a Cat-operad, are exactly the braided monoidal categories. Also, by work of \textit{F. Cohen} and \textit{P. Deligne}, the homology \(H_*(P_*)\) is the operad of Gerstenhaber algebras. The author uses the properties of \(H_*(P_n)\) to show that \(\pi_*((BP_n)_+\# K(k))\cong H_*(P_n)\otimes K_*(k)\), and so by means of the assembly map \((BP_n)_+\wedge K(k)\to K(k[P_n])\) we get the cited result. In the case of a balanced monoidal abelian category with exact tensor product the K-groups form a Batalin-Vilkovisky algebra. This algebraic structure is further analyzed in the semi-simple case, proving a generalization of a theorem of \textit{G. Anderson} and \textit{G. Moore} [Commun. Math. Phys. 117, 441-450 (1988; Zbl 0647.17012)] and \textit{G. Vafa} [Phys. Lett. B 206, 421-426 (1988)].