An example of a simple structurable algebra

Abstract

The class of structurable algebras has been defined by Allison [I] We denote by M the subspace of the tensor product S~q6 $ generated by the set IS t@$z -6~$/: $I,$26S }. We set ~=S~0S/~ and on the direct sum of the spaces ~S we define a commutative operation • and an anticormnutative operation [, ] by the rules:

Structurable superalgebras of Cartan type

“…(See B. Allison [1], O.N. Smirnov [16].) Let (A,¯) be a finite dimensional central simple structurable algebra over a field F of characteristic zero.…”

Gradings on tensor products of composition algebras and on the Smirnov algebra

“…Smirnov algebras are 35-dimensional simple exceptional structurable algebras. It is well-known ( [Smi90a]) that its derivation algebra is a simple Lie algebra of type G 2 , and its Kantor construction is a simple Lie algebra of type E 7 . We will recall now the definition of the Smirnov algebra.…”

“…In this construction, we always assume that C is a Cayley algebra over a field F of characteristic different from 2, with norm n and product · ; the bilinear form associated to the norm will be denoted by n too. We recall now the construction of the Smirnov algebra (see [Smi90a] for more details). Denote by S the 7-dimensional subspace of skew-symmetric elements of C and let [·, ·] be the commutator in C. Then, (S, [·, ·]) is a central non-Lie Malcev algebra, which is denoted by S (−) .…”

“…for any s, s i ∈ S and where the brackets [·, ·] on the right side of the equalities denote the product of S (−) . (Note that the third equation in (2.9) has a coefficient wrong in [Smi90a], which is corrected in [Smi92].) Then, the vector space T(C) := H ⊕ S with the new product (2.10)…”

Δ-Derivations OF SEMISIMPLE FINITE-DIMENSIONAL STRUCTURABLE ALGEBRAS

“…Let us give some examples of structurable algebras (see [21,22]): S1) Let (A,¯) be an unital associative algebra with the involution. Then (A,¯) is a structurable algebra.…”