Algebra and Applications 1. Abdenacer Makhlouf

Algebra and Applications 1

J is isomorphic to one of the superalgebras in examples 1.8, 1.9 and 1.10–1.15 or it is the Kantor double of the Poisson bracket in example 1.17.

REMARK 1.3.– We will assume always in this section that .

1.6.2. Case char F = p > 2, the even part is semisimple

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

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.

1.6.3. Case char F = p > 2, the even part is not semisimple

This case shows similarities with infinite dimensional superconformal Jordan algebras (see section 1.8) in characteristic 0.

Let us denote the algebra of truncated polynomials in m variables. Let B(m, n) = B(m) ⊗ G(n) be the tensor product of B(m) with the Grassmann algebra G(n) = 〈1, ξ₁,…, ξn〉. Then B(m, n) is an associative commutative superalgebra.

THEOREM 1.3 (Martínez and Zelmanov (2010)).– Let

be a finite dimensional simple unital Jordan superalgebra over an algebraically closed field F of characteristic p > 2. If the even part

is not semisimple, then there exist integers m, n and a Jordan bracket { , } on B(m, n) such that J = B(m, n) + B(m, n)v = KJ(B(m, n), { , }) is a Kantor double of B(m, n) or J is isomorphic to a Cheng–Kac Jordan superalgebra JCK(B(m), d) for some derivation d : B(m) → B(m).

1.6.4. Non-unital simple Jordan superalgebras

Finally, let us consider non-unital simple Jordan superalgebras. As we have seen, K₃ the three-dimensional Kaplansky superalgebra and K₉ the nine-dimensional degenerate Kac superalgebra are examples of such superalgebras.

EXAMPLE 1.23.– Let Z be a unital associative commutative algebra, D : Z → Z a derivation. Assume that Z is D-simple and that the only constants are elements α1, α ∈ F.

Let us consider in Z the bracket { , } given by:

The above bracket is a Jordan bracket, so the Kantor double V(Z, D) = Z + Zv = KJ(Z, { , }) is a simple

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