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.

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).

Abstract. We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.

Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real Clifford algebras in terms of the Arf invariant of the associated quadratic forms over the field of two elements, and give new proofs of some classical results.

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