Algebra & Number Theory

Constructing simply laced Lie algebras from extremal elements

Jan Draisma and Jos in ’t panhuis

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For any finite graph Γ and any field K of characteristic unequal to 2, we construct an algebraic variety X over K whose K-points parametrize K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the nonedges. After that, we study the case where Γ is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parametrize Lie algebras isomorphic to a single fixed Lie algebra. If Γ is of affine type, then this fixed Lie algebra is the split finite-dimensional simple Lie algebra corresponding to the associated finite-type Dynkin diagram. This gives a new construction of these Lie algebras, in which they come together with interesting degenerations, corresponding to points outside the open dense subset. Our results may prove useful for recognizing these Lie algebras.

Article information

Algebra Number Theory, Volume 2, Number 5 (2008), 551-572.

Received: 17 August 2007
Revised: 6 March 2008
Accepted: 27 May 2008
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 17B20: Simple, semisimple, reductive (super)algebras
Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B01: Identities, free Lie (super)algebras

Lie algebras extremal elements generators and relations


Draisma, Jan; in ’t panhuis, Jos. Constructing simply laced Lie algebras from extremal elements. Algebra Number Theory 2 (2008), no. 5, 551--572. doi:10.2140/ant.2008.2.551.

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