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QUIVER REPRESENTATIONS RESPECTING A QUIVER AUTOMORPHISM: A GENERALISATION OF A THEOREM OF KAC

Published online by Cambridge University Press:  28 January 2004

ANDREW HUBERY
ANDREW HUBERY
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT ahubery@maths.leeds.ac.uk Institut für Mathematik, Universität Paderborn, D-33095 Paderborn, Germanyhubery@math.upb.de
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Abstract

A theorem of Kac on quiver representations states that the dimension vectors of indecomposable representations are precisely the positive roots of the associated symmetric Kac–Moody Lie algebra. Here this result is generalised to representations respecting an admissible quiver automorphism, and the positive roots of an associated symmetrisable Kac–Moody Lie algebra are obtained.

Also the relationship with species of valued quivers over finite fields is discussed. It is known that the number of isomorphism classes of indecomposable representations of a given dimension vector for a species is a polynomial in the size of the base field. It is shown that these polynomials are non-zero if and only if the dimension vector is a positive root of the corresponding symmetrisable Kac–Moody Lie algebra.

Keywords

16G2016S3517B67
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 69 , Issue 1 , February 2004 , pp. 79 - 96
Copyright
The London Mathematical Society 2004

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