Classification of division gradings on finite-dimensional simple real algebras

Abstract

We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give the classification, up to isomorphism, of (not necessarily division) gradings on such algebras.Linear algebra over the field of two elements plays an interesting role in the proofs.Date: December 22, 2015.2010 Mathematics Subject Classification. Primary 16W50; secondary 16K20, 16S35.

Classification of involutions on graded-division simple real algebras

“…In this section we introduce the necessary terminology concerning quadratic forms on certain abelian groups, mainly following [13,Section 4]. That article established a correspondence between isomorphism classes of division gradings and quadratic forms that are regular in the sense that usually appears in the literature.…”

“…In this paper, however, we deal with quadratic forms that satisfy less restrictive conditions of regularity. The notation here is congruent with [13], but note that now we do not require quadratic forms and nice maps to be defined on elementary abelian 2-groups. Definition 11.…”

“…is either RI or RI ⊕ RZ, and it is easy to show that Z is homogeneous and its degree f is an element of order 2 (see for example [13,Lemma 14]). Now the classification is obtained by repeating the arguments of [13, Theorems 15 and 16].…”

“…These gradings are classified in [13,Theorems 15 and 16]; there are four families of equivalence classes: (1-a), (1-b), (1-c) and (1-d). We keep the same notation, so let T be the support of Γ, and let β : T × T → {±1} be the alternating bicharacter given by the commutation relations.…”

“…As a prerequisite, we need to know the classification of division gradings on finite-dimensional simple real algebras (without involution). This classification has been done in [13], both up to isomorphism and up to equivalence, and independently in [6], up to equivalence (but note that one of the equivalence classes was overlooked). A classification up to equivalence has been obtained in [7] without assuming simplicity.…”