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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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