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Let A be a self-injective connected Nakayama algebra.what is the Loewy length of any indecomposable projective A-module?

flag	asked Feb 9 2012 at 16:24 paria 11●1

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I don't understand what you are asking, really. As Julian observes, this depends on the algebra... (and not on the particular indecomposable projective, by the way!) but: what kind of answer were you expecting? – Mariano Suárez-Alvarez Feb 9 2012 at 19:43

Julian Kuelshammer · Answer 1 · 2012-02-09 16:56:37Z UTC

The Loewy length can be arbitrary. See e.g. [Assem, Simson, Skowronski: Elements of the Representation Theory of Associative Algebras 1] Proposition V.3.8:

$A$ basic, self-injective, connected Nakayama, then $A\cong kQ/I$, where $Q$ is an oriented cycle and $I=(kQ^+)^h$, where $(kQ^+)$ is the ideal spanned by all the arrows.

The Loewy length of each projective indecomposable should then be $h$.

This certainly also holds true for non-basic algebras if you replace isomorphism by Morita equivalence.

nakayama algebra

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