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

Abstract

We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.

arXiv comment61 pages. To appear in J. Algebra

References in corpus (9)

• Classification of group gradings on simple Lie algebras of types A, B, C and D
• Classification of division gradings on finite-dimensional simple real algebras
• Graded modules over classical simple Lie algebras with a grading
• Gradings on the Lie algebra $D_4$ revisited
• Graded polynomial identities as identities of universal algebras
• Gradings on classical central simple real Lie algebras
• Gradings on the simple real Lie algebras of types $G_2$ and $D_4$
• Classification of involutions on graded-division simple real algebras
• Fine gradings and automorphism groups on associative algebras