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Gaudin subalgebras and stable rational curves

Part of:Lie algebras and Lie superalgebrasRepresentation theory of groupsCurves

Published online by Cambridge University Press:  26 April 2011

Leonardo Aguirre,
Giovanni Felder and
Alexander P. Veselov
Leonardo Aguirre
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland (email: leonardo.aguirre@math.ethz.ch)
Giovanni Felder
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland (email: giovanni.felder@math.ethz.ch)
Alexander P. Veselov
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UKMoscow State University, Moscow 119899, Russia (email: a.p.veselov@lboro.ac.uk)
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Abstract

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Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra $\Xmathfrak {t}_{\hspace *{.3pt}n}$. We show that Gaudin subalgebras form a variety isomorphic to the moduli space $\bar M_{0,n+1}$ of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of $\bar M_{0,n+1}$ in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over $\bar M_{0,n+1}$ is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of $\bar M_{0,n+1}$.

Keywords

Gaudin modelsKohno–Drinfeld Lie algebrasstable curvesJucys–Murphy elements

MSC classification

Secondary:14H10: Families, moduli (algebraic)14H70: Relationships with integrable systems17B99: None of the above, but in this section20C30: Representations of finite symmetric groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011, pp. 1463 - 1478
Copyright
Copyright © Foundation Compositio Mathematica 2011

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