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

Abstract

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 A 3 ∼ = D 3 , B 2 ∼ = C 2 , and D 2 . 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: A 3 = sl(4, C), C 2 = sp(4, C), D 2 = o(4, C). The inclusions sl(4, C) ⊃ sp(4, C) and sl(4, C) ⊃ 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).

Fine gradings on simple classical Lie algebras

“…An explicit and irredundant description was later given for some classical simple Lie algebras of small rank (see the nice survey [Svo08] and references therein). Such descriptions are also known for the octonions [Eld98], the exceptional simple Jordan algebra (or Albert algebra) [DM09] and for the exceptional simple Lie algebras of type G 2 [DM06], and independently [BT09], and of type F 4 [DM09].…”

Gradings on Finite-Dimensional Simple Lie Algebras

“…4) and also that the parametrization given for these subgroups is redundant: the same conjugacy class can appear many times. An explicit (and irredundant) description of the corresponding fine gradings was later given for some classical simple Lie algebras of small rank-see [39] and references therein. Such descriptions are also known for the octonions [23], the exceptional simple Jordan algebra (the Albert algebra) [20], and the exceptional Lie algebras of types G 2 [7,19] and F 4 [20]; they were also announced for D 4 [21,22] and E 6 (conference presentation).…”

“…For results concerning gradings on real Lie algebras the reader is referred to [29] and to the recent survey [39].…”

Gradings on the real form 𝔢6,−26

“…These maximal quasitori are also called MAD-groups (Maximal Abelian Diagonalizable). Unfortunately, the knowledge of the MAD-groups of Aut( L) for L a real Lie algebra, is not an equivalent problem to that one of classifying fine gradings on L up to equivalence [Sv,§4].…”