January 25, 2016
MONOMIAL ALGEBRAS
JAN SCHRÖER
Contents
1. Monomial algebras
2. Examples
References
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Let K be a field.
1. Monomial algebras
Let A = KQ/I be a K-algebra, where KQ is the path algebra of a quiver Q and
I is an admissible ideal in KQ.
Then A is a monomial algebra if I can be generated by a set of paths in Q.
It is an open problem to find a characterization of the class of monomial algebras
which is independent of generators and relations. We refer to [BG] for an attempt in
this direction. Maybe such a characterization does not exists, and maybe monomial
algebras are not a meaningful class of algebras, except that they are easy to handle
(concerning certain aspects, like the construction of projectives or of certain projective resolutions, etc), and that they are a commonly used test class for conjectures,
etc. Various important results on monomial algebras can be found in [ZH].
2. Examples
Here are some classes of monomial algebras:
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Tree algebras;
Path algebras of quivers;
Nakayama algebras (up to Morita equivalence);
String algebras.
References
[BG] M.J. Bardzell, E. Green, An invariant characterization of monomial algebras, Comm. Algebra
27 (1999), no. 5, 2331–2344.
[ZH] B. Zimmermann-Huisgen, Predicting syzygies over monomial relations algebras, Manuscripta
Math. 70 (1991), no. 2, 157–182.
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JAN SCHRÖER
Jan Schröer
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn
Germany
E-mail address: [email protected]