action of the even part on the odd part and the products of two elements, respectively, are given by the following multiplication tables:
where xi × i = 0, x1 × 2 = –x2 × 1 = x3, x1 × 3 = –x3 × 1 = x2, –x2 × 3 = x1 = x3 × 2.
The superalgebra JCK(Z, d) is simple if and only if Z is d-simple, that is, Z does not contain proper d-invariant ideals (see Martínez and Zelmanov (2010)).
Let us remark that for Z = ℂ[t, t–1] the above construction leads to the Cheng–Kac superconformal algebra, that is, CK(6) = TKK(JCK(6)), where
with
1.6. Finite dimensional simple Jordan superalgebras
1.6.1. Case F is algebraically closed and char F = 0
Let us assume now that F is algebraically closed and char F = 0. Kac derived the classification of finite dimensional simple Jordan F-superalgebras from his classification of simple finite dimensional Lie superalgebras via the Tits–Kantor–Koecher construction.
THEOREM 1.1 (see Kac (1977a) and Kantor (1990)).– Let
REMARK 1.3.– We will assume always in this section that
1.6.2. Case char F = p > 2, the even part
Let us assume next that char F = p > 2 and the even part
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
This superalgebra is simple.
EXAMPLE 1.22.– Let
The action of
Shestakov (1997) proved that B is an alternative superalgebra and has a natural involution ∗ given by (a + m)∗ = ā – m,
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
1.6.3. Case char F = p > 2, the even part
This case shows similarities with infinite dimensional
