Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

Author: Svobodová M.

Source: Czechoslovak Journal of Physics, Volume 54, Number 5, May 2004 , pp. 509-516(8)

Publisher: Springer

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Abstract:

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.

We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.

Keywords: fine gradings; Lie algebras

Document Type: Research article

DOI: http://dx.doi.org/10.1023/B:CJOP.0000024954.37957.7b

Affiliations: 1: Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2, 120 00, Czech Republic

Publication date: 2004-05-01