Algebra and Applications 1. Abdenacer Makhlouf

Algebra and Applications 1 - Abdenacer Makhlouf


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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 + a1I, it follows that a0, a1I. So every graded ideal I satisfies image.

      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.

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

      1 i) (a∗)∗ = a, ∀a ∈ A;

      2 ii) .

      If ∗ : AA 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, ∗)= {aA | a = –a} is a Lie subsuperalgebra of A(–).

      The following two subsuperalgebras of image are of this type.

      Let us denote image.

      Then U t = U–1 = –U, and ∗ : Mn+2m(F) → Mn+2m(F) given by

image

      is a superinvolution.

      The superalgebras

image

      are the Lie and Jordan orthosymplectic superalgebras, respectively.

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

image

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

image

      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

image

      The superalgebra D(t) is simple if t0. In the case t = –1, the superalgebra D(–1) is isomorphic to M1+1(F)(+).

image

      for arbitrary v, ωV.

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

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

image

      and any other product of two basic elements is 0.

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

image

      Finally, the action of image over image is given by:

image

      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).

      DEFINITION


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