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

 

Algebras whose Tits form accepts a maximal omnipresent root


Authors: José A. de la Peña and Andrzej Skowroński
Journal: Trans. Amer. Math. Soc. 366 (2014), 395-417
MSC (2010): Primary 16G20, 16G60, 16G70
Published electronically: September 16, 2013
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Abstract: Let $ k$ be an algebraically closed field and $ A$ be a finite-dimensional associative basic $ k$-algebra of the form $ A=kQ/I$ where $ Q$ is a quiver without oriented cycles or double arrows and $ I$ is an admissible ideal of $ kQ$. We consider roots of the Tits form $ q_A$, in particular in the case where $ q_A$ is weakly non-negative. We prove that for any maximal omnipresent root $ v$ of $ q_A$, there exists an indecomposable $ A$-module $ X$ such that v=dim X. Moreover, if $ A$ is strongly simply connected, the existence of a maximal omnipresent root of $ q_A$ implies that $ A$ is tame of tilted type.


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Additional Information

José A. de la Peña
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, México 04510 D.F., México — and — Centro de Investigación en Matemáticas AC, La Valenciana, Guanajuato 36240 Gto. México
Email: jap@matem.unam.mx, jap@cimat.mx

Andrzej Skowroński
Affiliation: Faculty for Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: skowron@mat.uni.torun.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05841-2
PII: S 0002-9947(2013)05841-2
Keywords: Root of the Tits form, weakly non-negative quadratic form, maximal omnipresent root, tilted algebra, tame algebra, exceptional index
Received by editor(s): August 13, 2011
Received by editor(s) in revised form: March 28, 2012
Published electronically: September 16, 2013
Additional Notes: The work for this paper started during the visit of the second author to the Instituto de Matemáticas at UNAM during the spring of 2009.
Both authors acknowledge support from the Consejo Nacional de Ciencia y Tecnología of Mexico.
The second author has also been supported by the Research Grant No. N N201 269135 of the Polish Ministry of Science and Higher Education
Article copyright: © Copyright 2013 American Mathematical Society