Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 039, 13 pages      arXiv:0804.2209      https://doi.org/10.3842/SIGMA.2008.039
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms

Milena Svobodová
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13, 120 00, Praha 2, Czech Republic

Received August 31, 2007, in final form April 07, 2008; Published online April 14, 2008

Abstract
In this review paper, we treat the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions or deformations of the algebras. One basic question tackled in the work is the relation between the terms 'grading' and 'group grading'. Although these terms have originally been claimed to coincide for simple Lie algebras, it was revealed later that the proof of this assertion was incorrect. Therefore, the crucial statements about one-to-one correspondence between fine gradings and MAD-groups had to be revised and re-formulated for fine group gradings instead. However, there is still a hypothesis that the terms 'grading' and 'group grading' coincide for simple complex Lie algebras. We use the MAD-groups as the main tool for finding fine group gradings of the complex Lie algebras A3 @ D3, B2 @ C2, and D2. Besides, we develop also other methods for finding the fine (group) gradings. They are useful especially for the real forms of the complex algebras, on which they deliver richer results than the MAD-groups. Systematic use is made of the faithful representations of the three Lie algebras by 4 × 4 matrices: A3 = sl(4,C), C2 = sp(4,C), D2 = o(4,C). The inclusions sl(4,C) É sp(4,C) and sl(4,C) É o(4,C) are important in our presentation, since they allow to employ one of the methods which considerably simplifies the calculations when finding the fine group gradings of the subalgebras sp(4,C) and o(4,C).

Key words: Lie algebra; real form; MAD-group; automorphism; grading; group grading; fine grading.

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