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 A 3 over the complex number field. One of them ͑root decomposition͒ is the main tool of the theory and applications in working with A 3 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 A 3 is given in terms of the four-dimensional ͓sl͑4, C͔͒ and six-dimensional orthogonal ͓o͑6, C͔͒ 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.

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

“…These (mutually isomorphic) algebras are 10-dimensional. There are three non-conjugate MADgroups on the complex algebras, and thus three non-equivalent fine group gradings, which were found by the 'MAD-group' method in [19] and then confirmed for C 2 = sp(4, C) also by the 'displayed' method in [20] (where several representations with different defining matrices K of sp K (4, C) appear).…”

“…This statement holds for all finite-dimensional (not only simple) complex Lie algebras. The results in [19,20,21] thus remain valid when we replace the term fine grading by fine group grading.…”

“…We deal in fact again with two algebras that are isomorphic, namely A 3 = sl(4, C) and D 3 = o(6, C). The complex algebra has eight non-conjugate MAD-groups, and thus eight non-equivalent fine group gradings [20].…”

“…Therefrom, the crucial consequence was derived that for description of all fine gradings of a finite-dimensional simple complex Lie algebra it is sufficient to classify all the MAD-groups, which was done in the article [10] in 1998. On the basis of this classification, fine gradings for several low-rank Lie algebras were found [19,20,21].…”

“…In reaction to this revelation, the results in our articles [19,20,21] need to be revised. We have proved a statement about one-to-one correspondence between the MAD-groups and the fine group gradings.…”