Gradings on classical central simple real Lie algebras

Abstract

For any abelian group G, we classify up to isomorphism all Ggradings on the classical central simple Lie algebras, except those of type D 4 , over the field of real numbers (or any real closed field).

Classification of involutions on graded-division simple real algebras

“…In fact, we have already finished the classification of gradings on classical central simple real Lie algebras (except those of type D 4 ). The results are to appear in a separate article (see preprint [3]), in which some of the arguments rely on this paper. On the other hand, the classification of involutions (and related objects) may be of independent interest.…”

“…As mentioned in the Introduction, our purpose is to apply these results for the classification of gradings on classical real Lie algebras in a forthcoming article [3], so here we restrict ourselves to the situation relevant for that application.…”

“…As mentioned above, we use the results of this paper to classify gradings on classical real Lie algebras in [3]. There, in the case of outer gradings on special linear Lie algebras (which belong to series A), we have to deal with associative algebras that are not simple, but simple as algebras with involution.…”

“…We use them in our preprint [3], but they may also be of independent interest. For example, in the situation of Section 8, they allow us to construct a special basis for a part of the graded-division algebra.…”

Group gradings on the Lie and Jordan algebras of block-triangular matrices

“…(Only Type I exists for n = 2.) Their isomorphism classes are stated in Theorem 3.53 of [7], but we will use Theorem 45 of [2], which is equivalent but uses more convenient parameters.…”