Gradings on the simple real Lie algebras of types G2 and D4

Abstract

We classify group gradings on the simple Lie algebras of types G 2 and D 4 over the field of real numbers (or any real closed field): fine gradings up to equivalence and G-gradings, with a fixed group G, up to isomorphism.

Gradings on associative algebras with involution and real forms of classical simple Lie algebras

“…Finally, in Section 6, we review the transfer technique mentioned above and obtain, as a direct consequence of our results on associative algebras with involution, a classification of fine gradings up to equivalence for all real forms of classical simple Lie algebras except D 4 : see Theorems 6.6, 6.7, 6.8 and 6.10 for series A, Theorem 6.11 for series B (which is much easier than other series and could be treated with simpler techniques), Theorem 6.14 for Series C and Theorem 6.16 for series D. In the final subsection we complete the classification of fine gradings on real forms of type D 4 . We combine our results on associative algebras with involution and the results of [EK18] to classify what we call Type I and II fine gradings, to complement the classification of Type III fine gradings already obtained in [EK18].…”

“…This transfer was used in [BKR18b] to classify up to isomorphism all G-gradings on the real forms of classical simple Lie algebras except D 4 ; this work relies on the study of involutions on real graded-division algebras carried out in [BKR18a]. Finally, a classification of G-gradings on real forms of D 4 was obtained in [EK18].…”

On nonassociative graded-simple algebras over the field of real numbers

“…The following is a stronger version of the main result of [11], where it was used to classify gradings on Cayley algebras over algebraically closed fields (see also [13,Theorem 4.12]). Starting from this point, a classification of gradings up to isomorphism is obtained in [15] for any Cayley algebra. Here we will state the result only for F = R and F = C. To be consistent with our previous notation, we will denote the grading group by G.…”

“…(Here, as in the case of simple Lie algebras, there is no loss of generality in assuming the grading group abelian because the elements of the support always commute.) A classification of gradings up to isomorphism is given in [15]. We will use our extension of the loop algebra construction to derive a classification up to isomorphism of graded-simple alternative algebras that are graded-central over R and not associative (Theorem 4.4).…”

“…[15]). Any nontrivial grading on a Cayley algebra is, up to equivalence, either a grading induced by the Cayley-Dickson doubling process starting with a Hurwitz division subalgebra, or a coarsening of the Cartan grading on the split Cayley algebra.…”