Semigroup algebras of linear semigroups
The authors state that in this paper they begin a systematic study of the semigroup algebras of linear semigroups, i.e. the subsemigroups of the multiplicative monoids \({\mathcal M}_ n(k)\) of all \(n\times n\)-matrices over a field \(k\), which is supposed to be algebraically closed. \({\mathcal M}_ n(k)\) is considered with respect to its Zariski topology. If \(F\) is a field, \(F[S]\) denotes the semigroup algebra of a semigroup \(S\). First, the connections between \(F[S]\) and \(F[\overline S]\), where \(S\subset {\mathcal M}_ m(k)\) and \(\overline S\) is its (Zariski) closure, are established, and then only the case is considered, when \(S =\overline S\) is a connected monoid. [See \textit{M. S. Putcha}, ``Linear Algebraic Monoids'' (Lond. Math. Soc. Lect. Note Ser. 133, 1988; Zbl 0647.20066).] It is shown, that if \(F\) is any field of characteristic zero and \(M\) is a closed connected monoid, then the Jacobson radical \(J(F[M])\) is the kernel of the homomorphism \(F[M] \to F[M/\sim]\) for some congruence \(\sim\) on \(M\). If \(M\) is a connected monoid with zero, and \(F\) is any field, then \(F[M]\) is semisimple iff \(M\) is regular and the characteristic of \(F\) is either zero or else equals the characteristic of \(K\). There are more interesting results. Finally it is proved that if \(R\) is a unitary algebra over a field \(k\), and \(R^*\) denotes the multiplicative semigroup of \(R\), then for any field \(F\) of characteristic zero, \(F[R^*]\) is semisimple iff \(R\) is semiprime.
scientific article; zbMATH DE number 165839
scientific article