Algebra and Applications 1. Abdenacer Makhlouf

map { , } : A × A → A is called a Poisson bracket if

      1 1) (A, { , }) is a Lie superalgebra;

      2 2) {ab, c} = a{b, c} + (–1)|b||c|{a, c}b for arbitrary .

      EXAMPLE 1.16.– Let F[p₁,…, pn, q₁,…, qn] be a polynomial algebra in 2n variables. The classical Hamiltonian bracket:

      is a Poisson bracket.

EXAMPLE 1.17.– Let

      be the Grassmann algebra over an n-dimensional vector space V. Then the bracket

      for arbitrary is a Poisson bracket.

      DEFINITION 1.11.– Given

an associative commutative superalgebra, a bilinear map { , } : A × A → A is called a contact bracket if:

      1 i) (A, { , }) is a Lie superalgebra;

      2 ii) {ab, c} = a{b, c} + (–1)|b||c|{a, c}b + abD(c) for arbitrary homogeneous elements a, b, c in A.

      Note that a Poisson bracket is a contact bracket with D = 0.

      EXAMPLE 1.18.– Let F[t] be the polynomial algebra. Then the bracket {f(t), g(t)} = f′(t)g(t) – f (t)g′(t) is a contact bracket.

EXAMPLE 1.19.– Let Λ(1 : n) be the polynomial superalgebra in one (even) Laurent variable and n (odd) Grassmann variables ξ₁,…, ξn, Λ(1 : n) = F[t, t^–1, ξ₁,…, ξn]. Consider or a derivation of F[t, t^–1]. The bilinear map defined on generators of Λ(1 : n) by:

      can be extended to a contact bracket on Λ(1 : n).

      DEFINITION 1.12.– [Kantor double] Let A be an associative commutative superalgebra with a bilinear map { , } : A × A → A. Assume that . Consider a direct sum of vector spaces KJ(A, { , }) = A + Av where |v| = 1. Define a new product in J(A, { , }) that coincides with the original one in A and is given by:

      The superalgebra KJ(A, { , }) is called the Kantor double of (A, { , }).

      Kantor (1990) proved that if the bracket { , } is a Poisson bracket, then KJ(A, { , }) is a Jordan superalgebra.

      DEFINITION 1.13.– The bilinear map { , } is called a Jordan bracket on the superalgebra A if KJ(A, { , }) is a Jordan superalgebra (see King and McCrimmon (1992)).

      Cantarini and Kac (2007) showed that there is a 1-1 correspondence between Jordan brackets and contact brackets. Indeed, if [a, b] is a contact bracket with derivation D, D(a) = [a, 1], then the new bracket

      is a Jordan bracket.

Applying this to example 1.19, we get the following:

EXAMPLE 1.20.– The values {ξi, t} = 0, {ξi, ξj} = δij for 1 ≤ i, j ≤ n extend to a Jordan bracket of Λ(1 : n). Applying the Kantor double process to this bracket, we get Jordan superalgebras Jn = KJ(Λ(1 : n), { , }).

1.5. Cheng–Kac superalgebras

      Given an arbitrary unital associative commutative (super)-algebra with an (even) derivation d : Z → Z, Martínez and Zelmanov constructed (Martínez and Zelmanov 2010) a Jordan superalgebra JCK(Z, d) named the Cheng–Kac Jordan superalgebra.

      The even part of JCK(Z, d) is a rank 4 free module over Z with basis {1, w₁, w₂, w₃}, , and multiplication given by wiwj = 0 if 1 ≤ i ≠ j ≤ 3, . The odd part of this superalgebra is also a rank 4 free module over Z with basis {x, x₁, x₂, x₃}, .

      The 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, x_{1 × 2} = –x_{2 × 1} = x₃, x_{1 × 3} = –x_{3 × 1} = x₂, –x_{2 × 3} = x₁ = x_{3 × 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 (see section 1.8).

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

be a simple Jordan superalgebra over an