On the Fine Gradings of Simple Classical Lie Algebras

Abstract

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).