The eight fine gradings of sl(4, C) and o(6, C)

Abstract

A grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are eight fine gradings of the simple Lie algebra of type A3 over the complex number field. One of them (root decomposition) is the main tool of the theory and applications in working with A3 and with its representations; one other has also been used in the literature, and the rest have apparently not been recognized so far. An explicit description of all the fine gradings of A3 is given in terms of the four-dimensional [sl(4, ℂ)] and six-dimensional orthogonal [o(6, ℂ)] representations of the algebra. These results should be useful generally for choosing bases which reflect structural properties of the Lie algebra, for defining various sets of additive quantum numbers for systems with such symmetries, and for systematic study of grading preserving contractions of this Lie algebra.