Simple finite-dimensional modular noncommutative Jordan superalgebras

Abstract

We classify the central simple finite-dimensional noncommutative Jordan superalgebras of degree > 1 over an algebraically closed field of characteristic p > 2. The case of characteristic 0 was considered by the authors in the previous paper [24]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra B(m, n).

The universal conservative superalgebra

“…Comparing these expressions and using (10), we infer that ad(a)(α, β) ∈ J. The correctness of (24) and (25) follows from (13) and (14).…”

“…Now, by the formulas above this homomorphism in composition with the homomorphism which was constructed above is exactly the adjoint mapping. Now we show that ker(ad) = I, where I is the maximal Jacobi ideal of M. Indeed, if a ∈ I, then it follows from (13) and (14) that ad(a) = 0.…”

“…A superalgebra U is called a noncommutative Jordan superalgebra if U is flexible (that is, the operator identity [R x , L y ] = [L x , R y ] holds in U) and its symmetrized superalgebra (the algebra on the space of U with the multiplication x • y = 1 2 (xy + (−1) xy yx)) is Jordan. For more information on noncommutative Jordan superalgebras see [13,12] and references therein. Proof.…”

Representations of simple noncommutative Jordan superalgebras I

“…Their descriptions can be found in the paper [29]. Note also that a lot of simple Jordan superalgebras, such as P (2), K 10 , K 9 , do not admit nonzero Poisson brackets and do not give new examples of simple algebras (see [29], [30]).…”

“…The superalgebras K 10 and K 9 do not admit nonzero generic Poisson brackets (see [30]). In this section we classify noncommutative Jordan representations over K 10 and K 9 .…”