Algebra and Applications 1. Abdenacer Makhlouf

Algebra and Applications 1 - Abdenacer Makhlouf


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map { , } : A × AA 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:

image

      is a Poisson bracket.

image

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

image

      for arbitrary image is a Poisson bracket.

      DEFINITION 1.11.– Given

an associative commutative superalgebra, a bilinear map { , } : A × AA 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.

image

      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 × AA. Assume that image. 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:

image

      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

image

      is a Jordan bracket.

      Given an arbitrary unital associative commutative (super)-algebra with an (even) derivation d : ZZ, 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}, image, and multiplication given by wiwj = 0 if 1 ≤ ij ≤ 3, image. The odd part of this superalgebra is also a rank 4 free module over Z with basis {x, x1, x2, x3}, image.

      The action of the even part on the odd part and the products of two elements, respectively, are given by the following multiplication tables:

image

      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

image

      with image (see section 1.8).

      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.

be a simple Jordan superalgebra over an
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