The Lie algebra associated to a unit form

To any unit form q ( x ) = ∑ i = 1 n x i 2 + ∑ i < j q i j x i x j , q i j ∈ Z , we associate a Lie algebra G ˜ ( q ) —an intersection matrix Lie algebra in the terminology of Slodowy—by means of generalized Serre relations. For a nonnegative unit form the isomorphism type of G ˜ ( q ) is determi...

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Bibliographic Details
Published in:Journal of algebra 2006-02, Vol.296 (1), p.1-17
Main Authors: Barot, M., Kussin, D., Lenzing, H.
Format: Article
Language:English
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Summary:To any unit form q ( x ) = ∑ i = 1 n x i 2 + ∑ i < j q i j x i x j , q i j ∈ Z , we associate a Lie algebra G ˜ ( q ) —an intersection matrix Lie algebra in the terminology of Slodowy—by means of generalized Serre relations. For a nonnegative unit form the isomorphism type of G ˜ ( q ) is determined by the equivalence class of q. Moreover for q nonnegative and connected with radical of rank zero or one respectively, the algebras G ˜ ( q ) turn out to be exactly the simply-laced Lie algebras which are finite-dimensional simple or affine Kac–Moody, respectively. In case q is connected, nonnegative of corank two and not of Dynkin type A n , the algebra G ( q ) is elliptic.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2005.11.017