Fine Gradings Research Articles

On the Fine Gradings of Simple Classical Lie Algebras

Using the classification of maximal commutative subgroups in group of automorphisms of gl(n,C), we describe procedure for obtaining fine gradings in simple classical Lie algebras. This algorithm is applied to the algebras sl(4,C), o(4,C) and sp(4,C). Every grading is associated with label graphs. In this way, the problem whether two gradings are equivalent is reduced to the problem whether corresponding graphs are isomorphic. It is shown that there exist exactly 9 nonequivalent fine gradings in gl(4,C), 6 fine gradings in o(4,C) and 3 fine gradings in sp(4,C).

The four sets of additive quantum numbers of SU(3)

The four maximal sets of additive quantum numbers and related fine gradings of the Lie algebra sl(3,C) are described in detail. The quantum numbers are determined by a grading of the Lie algebra, fine gradings providing maximal sets of them. Two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the Lie algebra.