Path algebras of quivers and representations of locally finite Lie algebras

We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras \(\mathfrak{sl}(n^\infty)\), \(\mathfrak o(n^\infty)\), and \(\mathfrak{sp}(n^\infty)\). Let \(\mathfrak g_\infty\) be one of these Lie algebras, and let \(I \subseteq U(\mathfrak g...

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Bibliographic Details
Published in:arXiv.org 2016-06
Main Authors: Hennig, J, Sierra, S J
Format: Article
Language:English
Subjects:
Algebra
Lie groups
Modules
Representations
Online Access:Get full text
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Summary:We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras \(\mathfrak{sl}(n^\infty)\), \(\mathfrak o(n^\infty)\), and \(\mathfrak{sp}(n^\infty)\). Let \(\mathfrak g_\infty\) be one of these Lie algebras, and let \(I \subseteq U(\mathfrak g_\infty)\) be the nonzero annihilator of a locally simple \(\mathfrak g_\infty\)-module. We show that for each such \(I\), there is a quiver \(Q\) so that locally simple \(\mathfrak g_\infty\)-modules with annihilator \(I\) are parameterised by "points" in the "noncommutative space" corresponding to the path algebra of \(Q\). Methods of noncommutative algebraic geometry are key to this correspondence. We classify the quivers that arise and relate them to characters of symmetric groups.
ISSN:2331-8422