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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Product systems over right-angled Artin semigroups


Authors: Neal J. Fowler and Aidan Sims
Journal: Trans. Amer. Math. Soc. 354 (2002), 1487-1509
MSC (1991): Primary 20F36; Secondary 18B40, 55N20
Published electronically: November 30, 2001
MathSciNet review: 1873016
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Abstract: We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid $\mathcal G$. We show that the existing notions of product systems fit into our categorical framework, as do the $k$-graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid $\mathcal G$. In particular, we obtain necessary and sufficient conditions under which a collection of $k$ $1$-graphs form the coordinate graphs of a $k$-graph.


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Additional Information

Neal J. Fowler
Affiliation: Department of Mathematics, University of Newcastle, NSW 2308, Australia

Aidan Sims
Affiliation: Department of Mathematics, University of Newcastle, NSW 2308, Australia

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02911-7
PII: S 0002-9947(01)02911-7
Received by editor(s): December 22, 1999
Received by editor(s) in revised form: June 28, 2001
Published electronically: November 30, 2001
Article copyright: © Copyright 2001 American Mathematical Society