A class of nonassociative algebras with involution containing the class of Jordan algebras

Abstract

If ~' is a Jordan algebra, then Str(s¢) = L~O Der(s~¢) forms a Lie algebra called the structure algebra of d, where L, denotes the left multiplication operator by a for ae ~4 and Der(.~) is the algebra of derivations of J. The Tits-Koecher construction gives the vector space ~(~Str(~c)Gd the structure of a Lie algebra, where s] is a second copy of ~¢ (see Section 8.5 of [6]). In this paper, we define and study a cla~s of non-associative algebras containing the class of Jordan algebras and allowing the construction of generalizations of the structure algebra and the Tits-Koecher algebra. The algebras in this class, called structurable algebras, are unital algebras with involution. The class is defined by an identity of degree 4 and includes associative algebras, Jordan algebras (with the identity map as involution), tensor products of two composition algebras, the 56-dimensional Freudenthal module for E; with a natural binary product, and algebras constructed from hermitian forms in a manner generalizing the usual construction of Jordan algebras from quadratic forms (see Section 8). Throughout the paper, motivation for definitions and proofs is provided by the examples of Jordan algebras and composition algebras.The present paper is devoted primarily to the study of finite dimensional central simple structurable algebras over a field 4} of characteristic zero. The main result (Theorem 25) gives a complete description of these algebras. The primary toot used to obtain this description is the form defined by (x,y)= tr(Lx~+~.~), where -denotes the involution (see Section 6). A subsequent paper will deal with the generalization of the Tits-Koecher construction. The Lie algebras obtained from central simple structurable algebras using this construction include all finite dimensional central simple Lie algebras over 4} that contain at least one non-zero ad-nilpotent element.Constructions analogous to the Tits-Koecher construction have been studied in [1,3,4,8] and [16]. The algebras used to construct Lie algebras in these papers are ternary algebras. The simple Lie algebras obtained from these ternary algebras can be obtained from structurable algebras [although the graded structure obtained may differ in the two cases]. The formulation of the constructions in terms of binary algebras rather than ternary algebras results in simplification not only of

Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence

“…The XJCCorrespondence assures that we can associate to f = [F ] ∈ Bir 2,2 (P n−1 ) an irreducible projective variety X f ⊂ P 2n+1 , of dimension n, defined as the closure of the image of the affine parametrization x → [1 : (6) Let J be a rank 3 Jordan algebra with trace T (x) and cubic norm N (x). By [1,Section 8.v], the space of Zorn matrices with coefficients in J defined by…”

A Characterization of (−1, −1)-Freudenthal–kantor Triple Systems

“…Within the general framework of (ε, δ)-FKTSs (ε, δ = ±1) and the standard embedding Lie (super)algebra construction studied in [6, 7, 14-16, 27], we define δ-structurable algebras as a class of nonassociative algebras with involution which coincides with the class of structurable algebras for δ = 1 as introduced and studied in [1,2]. Structurable algebras are a class of nonassociative algebras with involution that include Jordan algebras (with trivial involution), associative algebras with involution, and alternative algebras with involution.…”

A Structure Theory of (−1,−1)-Freudenthal Kantor Triple Systems

“…Hence within the general framework of ( , δ)-FKTSs ( , δ = ±1) and the standard embedding Lie (super)algebra construction studied in [6,7,[13][14][15]28] (see also references therein) we define δ-structurable algebras as a class of nonassociative algebras with involution which coincides with the class of structurable algebras for δ = 1 as introduced and studied in [1,2]. Structurable algebras are a class of nonassociative algebras with involution that include Jordan algebras (with trivial involution), associative algebras with involution, and alternative algebras with involution.…”

“…The identity element of A is denoted by 1. Since char = 2, by [1] we have A = H ⊕ S, where H = {a ∈ A|a = a} and S = {a ∈ A|a = −a}.…”

“…By Remark (iii) above T A ∩ Der(A, − ) = 0 and so we may define the structure algebra Str(A, − ) := T A ⊕ Der(A, − ). This algebra plays an important role in the structure study of structurable algebras [1] and may play a role in the structure study of anti-structurable algebras (theory to be presented elsewhere).…”