Jordan quadruple systems

Abstract

Abstract. We define Jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the Jordan tetrad {a, b, c, d} = abcd + dcba as a quadrilinear operation on associative algebras. We find further identities in degree 10 which are not consequences of the defining identities. We introduce four infinite families of finite dimensional Jordan quadruple systems, and construct the universal associative envelope for a small system in each family. We obtain analogous results for the anti-tetrad [a, b, c, d] = abcd − dcba. Our methods rely on computer algebra, especially linear algebra on large matrices, the LLL algorithm for lattice basis reduction, representation theory of the symmetric group, noncommutative Gröbner bases, and Wedderburn decompositions of associative algebras. IntroductionIn this paper we study the quadrilinear operations {a, b, c, d} = abcd + dcba and [a, b, c, d] = abcd − dcba in associative algebras. The first is the Jordan tetrad which plays an important role in the structure theory of Jordan algebras [28,31]. The second is the anti-tetrad, which seems not to have been studied until now.1.1. Motivation. In an associative algebra for n ≥ 2 we define the n-tad to be this n-ary multilinear operation: {a 1 , . . . , a n } = a 1 · · · a n + a n · · · a 1 . For n = 2 we obtain the Jordan product {a, b} = ab+ba satisfying commutativity and the Jordan identity: a}, {{{a, a}, b}, a} ≡ {{a, a}, {b, a}}. There are further "special" identities satisfied by the Jordan product in every associative algebra which do not follow from the defining identities; the simplest occur in degrees 8 and 9 and are called the Glennie identities [17,18]. A Jordan algebra is "special" if it can be represented as a subspace of an associative algebra closed under the Jordan product; otherwise, it is "exceptional". If a special Jordan algebra is finite dimensional then its universal associative enveloping algebra is also finite dimensional. For n = 3 we obtain the Jordan triple product abc + cba; in every associative algebra, this operation satisfies identities which define Jordan triple systems (JTS): b, a}, {{a, b, c}, d, e} ≡ {{a, d, e}, b, c} − {a, {b, e, d}, c} + {a, b, {c, d, e}}. In contrast to the Jordan identity these identities are multilinear. There are special identities in higher degree: identities satisfied by the Jordan triple product in every associative algebra but which do not follow from the defining identities [24,25]. For the classification of finite dimensional JTS, see [22,29,30] and for their universal associative envelopes, see [23]. Closely related to Jordan triple systems are the anti-Jordan triple systems (AJTS), see [16]. Finite dimensional simple AJTS have been classified [1]. These systems are defined by identities satisfied by the anti-Jordan triple product abc−cba in every associative algebra:Universal associative envelopes for one infinite family of simple AJTS have been constructed [15]. At the next step n = 4 we obtain the Jordan tetrad abcd + dcba, which arises in...

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