Algebra and Applications 1. Abdenacer Makhlouf

is simple if it does not have non-trivial graded ideals. A graded ideal is an ideal I ⊴ A such that for every a = a₀ + a₁ ∈ I, it follows that a₀, a₁ ∈ I. So every graded ideal I satisfies .

      Wall (1963, 1964) proved that an arbitrary simple finite dimensional superalgebra over an algebraically closed field is isomorphic to one of the following two types:

      1 I) .

      2 II) .

      Consequently, we can easily get the first examples of simple finite dimensional Jordan superalgebras as explained above.

EXAMPLE 1.8.– .

EXAMPLE 1.9.– J = Q(n)⁽⁺⁾, n ≥ 2.

      DEFINITION 1.9.– Let A be an associative superalgebra. A map ∗ : A → A is a superinvolution if it satisfies:

      1 i) (a∗)∗ = a, ∀a ∈ A;

      2 ii) .

      If ∗ : A → A is a superinvolution of the associative superalgebra A, then the set of symmetric elements H(A, ∗) is a Jordan superalgebra of A⁽⁺⁾. Similarly, the subspace of skew-symmetric elements K(A, ∗)= {a ∈ A | a^∗ = –a} is a Lie subsuperalgebra of A^(–).

      The following two subsuperalgebras of are of this type.

EXAMPLE 1.10.– Let In, Imbe the identity matrices and let t be the transposition.

      Let us denote .

      Then U t = U^–1 = –U, and ∗ : Mn+_2m(F) → Mn+_2m(F) given by

      is a superinvolution.

      The superalgebras

      are the Lie and Jordan orthosymplectic superalgebras, respectively.

      EXAMPLE 1.11.– The associative superalgebra Mn+n(F) has another superinvolution given by:

      The Lie and Jordan superalgebras (respectively) that consist of skew-symmetric and symmetric elements, respectively, are denoted as P_n(F) and JP_n(F) (and are also called “strange series”).

EXAMPLE 1.12.– The three-dimensional Kaplansky superalgebra K₃= Fe + (Fx + Fy) with multiplication table:

      is a simple Jordan superalgebra. Note that K₃ is not unital.

      EXAMPLE 1.13.– The one-parametric family of four-dimensional superalgebras D(t) defined as D(t) = (Fe₁+ Fe₂) + (Fx + Fy) with the product

      The superalgebra D(t) is simple if t ≠ 0. In the case t = –1, the superalgebra D(–1) is isomorphic to M₁₊₁(F)⁽⁺⁾.

EXAMPLE 1.14.– Let V be a vector space that is ℤ/2ℤ-graded, , and has a superform (|) : V × V → F, which is symmetric on , skew-symmetric in and . Then is a Jordan superalgebra, where the product of two arbitrary elements α1 + v and β1 + ω in J is given by

      for arbitrary v, ω ∈ V.

      We will refer to J as the superalgebra of a superform.

      J is simple if and only if the form (|) is non-degenerate.

EXAMPLE 1.15.– Kac (1977a) introduced a 10-dimensional Jordan superalgebra J whose even part has dimension 6 and splits as the direct sum of a superalgebra of a superform and a one-dimensional algebra. Thus, has the multiplication given by:

      and any other product of two basic elements is 0.

      The odd part has a basis {x₁, x₂, y₁, y₂} and the following multiplication table:

      Finally, the action of over is given by:

      This superalgebra is simple if char F ≠ 3. In case of char F = 3, it has an ideal of dimension 9 that is spanned by e, vi, 1 ≤ i ≤ 4, xj, yj, 1 ≤ j ≤ 2. It is called degenerated Kac superalgebra and is denoted by K₉.

      In Medvedev and Zelmanov (1992), it is proved that the Kac superalgebra K₁₀ is not a homomorphic image of a special Jordan superalgebra.

      Benkart and Elduque (2002) realized K₁₀ as the space of K₃ ⊗ K₃ + F1 (with a new product) where K₃ is the Kaplansky superalgebra.

      Another (octonionic) construction of K₁₀ was suggested in Racine and Zelmanov (2015).

1.4. Brackets

      DEFINITION