TY - JOUR

T1 - Quivers of monoids with basic algebras

AU - Margolis, Stuart

AU - Steinberg, Benjamin

PY - 2012/9

Y1 - 2012/9

N2 - 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.

AB - 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.

KW - EI-categories

KW - Hochschild-Mitchell cohomology

KW - Monoids

KW - Quivers

KW - Representation theory

UR - http://www.scopus.com/inward/record.url?scp=84867115474&partnerID=8YFLogxK

U2 - 10.1112/S0010437X1200022X

DO - 10.1112/S0010437X1200022X

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AN - SCOPUS:84867115474

SN - 0010-437X

VL - 148

SP - 1516

EP - 1560

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 5

ER -