Читать книгу Algebra and Applications 1 - Abdenacer Makhlouf - Страница 17

Let us assume next that char F = p > 2 and the even part is a semisimple Jordan algebra.

Recall that a semisimple Jordan algebra is a direct sum of finitely many simple ideals.

This case was addressed in Racine and Zelmanov (2003) and the classification essentially coincides with the one of zero characteristic, expect of some differences if char F = 3.

EXAMPLE 1.21.– Let H₃(F), K₃(F) denote the symmetric and skew-symmetric 3×3 matrices over F, char F = 3. Consider and the sum of two copies of K₃(F). We have a Jordan superalgebra structure on via a ∙ b = a ∙ b in M3(F)⁺ if a, b ∈ H3(F), that is,

This superalgebra is simple.

EXAMPLE 1.22.– Let with , , where F is a field, char F = 3. The product in is given by:

The action of over is defined as follows:

Shestakov (1997) proved that B is an alternative superalgebra and has a natural involution ∗ given by (a + m)∗ = ā – m, , where a ↦ ā is the symplectic involution, and .

If H₃(B, ∗) denotes the symmetric matrices with respect to the involution ∗, then H₃(B, ∗) is a simple Jordan superalgebra. It is i-exceptional, that is, it is not a homomorphic image of a special Jordan superalgebra.

THEOREM 1.2 (Racine and Zelmanov (2003)).– Let be a finite dimensional central simple Jordan superalgebra over an algebraically closed field F of char F = p > 2. If and is semisimple, then J is isomorphic to one of the superalgebras in examples 1.8, 1.9, 1.10–1.14 or char F = 3 and J is the nine-dimensional degenerate Kac superalgebra (see example 1.15) or J is isomorphic to one of the superalgebras in examples 1.21 and 1.22.