A Groupoid Approach to Discrete Inverse Semigroup Algebras
A Groupoid Approach to Discrete Inverse Semigroup Algebras
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DOI：
10.1016/j.aim.2009.09.001
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发表时间：
2009-03
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通讯作者：
B. Steinberg
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作者：
B. Steinberg
设K是有单位的交换环，S是逆半群.我们证明了半群代数KS可以被描述为一个函数的卷积代数上的泛Etale广群相关联的S帕特森。这个结果同时推广了作者早期关于有限逆半群的工作和Paterson关于泛C-代数的定理。它提供了一个方便的拓扑框架来理解KS的结构，包括中心和当它有一个单位。在这个理论中，Gelfand对偶的作用被Stone对偶取代。利用这种方法，我们构造了任意域上逆半群的有限维不可约表示，并将其作为伴随群的诱导表示，推广了具有多个幂等元的逆半群的情形.更一般地说，我们描述了逆半群S的不可约表示，它可以从相关群中导出，恰好是那些满足一定“有限性条件”的表示。例如，其像包含本原幂等元的逆半群的所有表示都满足这个“有限性条件”。
Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C∗-algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition.” This “finiteness condition” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.