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Quivers of monoids with basic algebras

Part of: Representation theory of rings and algebras Semigroups Algebraic combinatorics

Published online by Cambridge University Press:  25 July 2012

Stuart Margolis  and
Benjamin Steinberg
Stuart Margolis
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel (email: margolis@math.biu.ac.il) Center for Algorithmic and Interactive Scientific Software, City College of New York, City University of New York, NY 10031, USA
Benjamin Steinberg
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada (email: bsteinberg@ccny.cuny.edu) Department of Mathematics, City College of New York, NY 10031, USA
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Abstract

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We compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of ℛ-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

Keywords

quiversrepresentation theorymonoidsHochschild–Mitchell cohomologyEI-categories

MSC classification

Secondary: 16G10: Representations of Artinian rings 20M25: Semigroup rings, multiplicative semigroups of rings 05E99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1516 - 1560
Copyright
Copyright © Foundation Compositio Mathematica 2012

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