Cellularity of twisted semigroup algebras.
In this paper the authors continue the investigations on cellularity of twisted semigroup algebras over an integral domain. The notion of cellular algebra was introduced by \textit{J. J. Graham} and \textit{G. I. Lehrer} [in Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)]. Many examples of such algebras have been found, including Hecke algebras of finite types, \(q\)-Schur algebras, Brauer algebras, Temperley-Lieb algebras, partition algebras, Birman-Wenzl algebras and many other diagram algebras. All these algebras give cellular semigroup algebras by specializing their defining parameters. However, it easily is seen that not all semigroup algebras are cellular. Hence two natural questions occur: (1) When is a semigroup algebra cellular? (2) How to get different cellular structures by deforming semigroup algebras? \textit{S. Wilcox} [in J. Algebra 309, No. 1, 10-31 (2007; Zbl 1154.16021)] considers the cellularity of twisted finite regular semigroup algebras over a commutative ring \(R\): a sufficient condition is given for such a ring to be cellular with respect to an involution which is determined by an involution on the semigroup itself and an involution of \(R\). This extends an earlier result of \textit{J. East} [J. Algebra 296, No. 2, 505-519 (2006; Zbl 1127.16021)] on cellularity of finite inverse semigroup algebras. In this paper the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of twisted semigroup algebras of type \(\mathcal{JH}\). Examples are partition algebras, Brauer algebras and Temperley-Lieb algebras. The main result obtained states that the twisted semigroup algebra of a regular semigroup algebra is cellular of type \(\mathcal{JH}\) with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular. Note that it is not assumed that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Furthermore, for a twisted semigroup algebra it is not required that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units of the ground ring. So, the results obtained in this paper extend those obtained by East, and some obtained by Wilcox.
scientific article; zbMATH DE number 5364857
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