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On the magnitude of a finite dimensional algebra

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Joseph Chuang, Alastair King and Tom Leinster

There is a general notion of the magnitude of an enriched category,
defined subject to hypotheses. In topological and geometric contexts,
magnitude is already known to be closely related to classical
invariants such as Euler characteristic and dimension. Here we
establish its significance in an algebraic context. Specifically, in
the representation theory of an associative algebra $A$, a central
role is played by the indecomposable projective $A$-modules, which
form a category enriched in vector spaces. We show that the magnitude
of that category is a known homological invariant of the algebra:
writing $\chi_A$ for the Euler form of $A$ and $S$ for the direct sum
of the simple $A$-modules, it is $\chi_A(S,S)$.

Keywords:
algebra, magnitude, indecomposable projective, simple module,
Cartan matrix, Euler form, Cartan determinant conjecture

2010 MSC:
18G99 (primary), 16D40, 16D60, 16G99, 18E05, 18G15

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 3, pp 63-72.

Published 2016-01-07.

http://www.tac.mta.ca/tac/volumes/31/3/31-03.pdf

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