Fine gradings on simple classical Lie algebras

Abstract

The fine abelian group gradings on the simple classical Lie algebras (including D 4 ) over algebraically closed fields of characteristic 0 are determined up to equivalence. This is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them.

Fine gradings on $\mathfrak{e}_6$

“…There are 7 MAD-groups of Aut c 4 , according to [24,18]. The induced fine gradings can also be extracted from [6], although in such paper there is one missing grading.…”

“…For algebraically closed fields of characteristic zero, the gradings on the classical Lie algebras have been studied in [6,9,16,18] and the gradings in some exceptional ones, namely, f 4 and g 2 , in [14,15,13,7]. Lately, some authors have already studied the case of prime characteristic, [5] in the classical case (with the exception of d 4 ) and [20] in f 4 and g 2 .…”

Counting Fine Gradings on Matrix Algebras and on Classical Simple Lie Algebras

“…If a group G acts on X, then it also acts on the multisets in X. The relevant group here is ASp 2m (2), the semidirect product Z Theorem 3 ( [Eld10,EK12c]). Let F be an algebraically closed field, char F = 2.…”

“…[EK13] and references therein). In particular, a classification of fine gradings up to equivalence is known for matrix algebras over an algebraically closed field F of arbitrary characteristic [HPP98a,BSZ01,BZ03] and for classical simple Lie algebras except D 4 in characteristic different from 2 [Eld10,EK12c]. Type D 4 is different from all other members of Series D due to the phenomenon of triality.…”

“…The more recent classification results, as stated in [EK12c] and [EK13], reduce the problem to counting orbits of certain finite groups, so the number of fine gradings can be computed, in principle, for any member of these series over an algebraically closed field of characteristic different from 2. Note that there is no work to be done for Series B because there are exactly r + 1 gradings on the simple Lie algebra of type B r (see [Eld10] or [EK13, §3.4]). We count the orbits using Burnside-CauchyFrobenius Lemma and the computer algebra system GAP (see [GAP]) to obtain the exact number of fine gradings for simple Lie algebras of types A r , C r and D r up to r = 100 (see Tables 4, 6 and 8, respectively).…”