Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities a(bc) = a(cb), (ba)a 2 = (ba 2 )a, (b, a 2 , c) = 2(b, a, c)a.These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤ 7 is a consequence of the three identities in degree ≤ 4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.where (a, b, c) = (ab)c − a(bc). The multilinear forms of the last two identities are J = (a(bc))d + (a(bd))c + (a(cd))b − (ab)(cd) − (ac)(bd) − (ad)(bc),Remark 9. Equivalent identities for the opposite product appear in the work of Kolesnikov. If we replace xy by yx in equations (26) and (27) of [17], replace x 1 , x 2 , x 3 , x 4 by a, b, c, d and apply right commutativity, then we obtain One easily verifies thatSee also the remarks on Jordan dialgebras in Pozhidaev [25], Section 3. Lemma 10. (Velásquez and Felipe [31]) The quasi-Jordan product in an associative dialgebra satisfies right commutativity and the quasi-Jordan identity.Lemma 11. (Bremner [3]) The quasi-Jordan product in an associative dialgebra satisfies the associator-derivation identity. The identities of Definition 8 imply every identity of degree ≤ 4 for the quasi-Jordan product in an associative dialgebra.Definition 12. A quasi-Jordan algebra is a nonassociative algebra over a field of characteristic = 2, 3 satisfying right commutativity and the quasi-Jordan identity.A semispecial quasi-Jordan algebra (also called a Jordan dialgebra) is a quasi-Jordan algebra satisfying the associator-derivation identity.Remark 13. Strictly speaking, a Jordan dialgebra as defined by Kolesnikov [17] and Pozhidaev [25] has two binary operations ⊣ and ⊢ which are in fact opposite as a result of the dialgebra version of commutativity, x ⊣ y = y ⊢ x. We simplify the notation by using only one operation and writing this operation as juxtaposition.Definition 14. A quasi-Jordan algebra is special if it is isomorphic to a subalgebra of D + for some associative dialgebra D. Every special algebra is semispecial.Glennie [7,8,9] (see also Hentzel [12]) discovered an identity satisfied by special Jordan algebras that is not satisfied by all Jordan algebras. In this paper we consider the analogous question for quasi-Jordan algebras. We use computer algebra to show that the identities in Definition 8 imply every identity of degree ≤ 7 for the quasi-Jordan product in an associative dialgebra. We demonstrate the existence of identities in degree 8 which do not follow from the identities of Definition 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity. These new identities are special identities in the following sense.
We compute the identities and the central identities of degree F6 of the Cayley᎐Dickson algebras. Our process can be used to compute identities of higher degree. We assume a field of characteristic 0 or greater than the degree of the identities studied. Our identities are homogeneous multilinear polynomials.ᮊ 1997 Academic Press
We determine structure constants for the universal nonassociative enveloping algebra U(M) of the fourdimensional non-Lie Malcev algebra M by constructing a representation of U(M) by differential operators on the polynomial algebra P (M). These structure constants involve Stirling numbers of the second kind. This work is based on the recent theorem of Pérez-Izquierdo and Shestakov which generalizes the Poincaré-Birkhoff-Witt theorem from Lie algebras to Malcev algebras. We use our results for U(M) to determine structure constants for the universal alternative enveloping algebra A(M) = U(M)/I(M) where I(M) is the alternator ideal of U(M). The structure constants for A(M) were obtained earlier by Shestakov using different methods.Keywords Keywords structure constant, four-dimensional malcev algebra, differential operator, recent theorem, different method, malcev algebra, poincar birkhoff-witt theorem, polynomial algebra, second kind, alternator ideal, lie algebra, four-dimensional non-lie malcev algebra Disciplines Disciplines Algebra | Mathematics Comments Comments This article is published as Bremner, Murray R., Irvin R. Hentzel, Luiz A. Peresi, and Hamid Usefi. "Universal enveloping algebras of the four-dimensional Malcev algebra." Contemporary Mathematics 483 (2009): 73-89. Posted with permission.Abstract. We determine structure constants for the universal nonassociative enveloping algebra U (M) of the four-dimensional non-Lie Malcev algebra M by constructing a representation of U (M) by differential operators on the polynomial algebra P (M). The structure constants for U (M) involve the Stirling numbers of the second kind. This work is based on the recent theorem of Pérez-Izquierdo and Shestakov which generalizes the Poincaré-Birkhoff-Witt theorem from Lie algebras to Malcev algebras. We use our results for U (M) to determine structure constants for the universal alternative enveloping algebra A(M) = U (M)/I(M) where I(M) is the alternator ideal of U (M). The structure constants for A(M) were obtained earlier by Shestakov using different methods.
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