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Full quivers of representations of algebras


Authors: Alexei Belov-Kanel, Louis H. Rowen and Uzi Vishne
Journal: Trans. Amer. Math. Soc. 364 (2012), 5525-5569
MSC (2010): Primary 16R99, 16G99
Published electronically: May 24, 2012
MathSciNet review: 2931338
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Abstract: We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to Zariski closed algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: Frobenius (along the diagonal) and, more generally, proportional Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.


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Additional Information

Alexei Belov-Kanel
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: belova@macs.biu.ac.il

Louis H. Rowen
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: rowen@macs.biu.ac.il

Uzi Vishne
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: vishne@macs.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2012-05565-6
Received by editor(s): May 31, 2010
Received by editor(s) in revised form: January 27, 2011
Published electronically: May 24, 2012
Additional Notes: This research was supported by the Israel Science Foundation (grant No. 1178/06).
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.