Hochschild cohomology algebra of abelian groups (Q1359057)

We present a direct proof of what is suggested by \textit{T. Holm}'s results [Commun. Algebra 24, No. 6, 1957-1969 (1996; Zbl 0853.16016)]: there is an isomorphism of algebras \(HH^*(kG,kG)\to kG\otimes H^*(G,k)\) where \(G\) is a finite abelian group, \(k\) a ring, \(HH^*(kG,kG)\) is the Hochschild cohomology algebra and \(H^*(G,k)\) the usual cohomology algebra. This result agrees with the well-known additive structure result in force for any group \(G\); we remark that the multiplicative structure result we have obtained is quite similar to the description of the monoidal category of Hopf bimodules over \(kG\) [given in \textit{C. Cibils}, Proc. Am. Math. Soc. 125, No. 5, 1315-1321 (1997; Zbl 0865.18005)]. This similarity leads to conjecture the structure of \(HH^*(kG,kG)\) for \(G\) a finite group.