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Bremner, Murray R.; Peresi, Luiz A.

Special identities for quasi-Jordan algebras. (English) Zbl 1241.17032

Commun. Algebra 39, No. 7, 2313-2337 (2011).
J.-L. Loday introduced introduced [C. R. Acad. Sci., Paris, Sér. I 321, No. 2, 141–146 (1995; Zbl 0845.16036); Lect. Notes Math. 1763, 7–66 (2001; Zbl 0999.17002)] a class of algebras with two associative operations satisfying a certain compatibility condition (the so called associative dialgebras). From these it is natural to generalize the idea to some other classes of nonassociative algebras. In an associative dialgebra \(A\) over a field of characteristic other than two and with products \(\vdash\) and \(\dashv\) we can define a new product \[ a\triangleleft b:={1\over 2}(a\vdash b+b\dashv a). \] This is the so called quasi-Jordan product in \(A\). It has been proved by R. Velásquez and R. Felipe [Commun. Algebra 36, No. 4, 1580–1602 (2008; Zbl 1188.17021)] that the quasi-Jordan product in an associative dialgebra satisfies the right commutativity identity \(a(bc)=a(cb)\) and the quasi-Jordan identity \((ba)a^2=(ba^2)a\). On the other hand M. R. Bremner [Commun. Algebra 38, No. 12, 4695–4704 (2010; Zbl 1241.17001)] proved that it also satisfies the associator-derivation identity \((b,a^2,c)=2(b,a,c)a\) where \((a,b,c)= (ab)c-a(bc)\).
In the work under review it is proved that the right commutativity, quasi-Jordan and associator-derivation identities imply every identity of degree \(\leq 7\) for the quasi-Jordan product in an associative dialgebra. It is also proved that there are identities of degree \(8\) which do not follow from the mentioned three identities. Some of these new identities are related to the Glennie identity. Among the techniques used in this research we would like to say that it includes computational algebra as well as representation theory.
Reviewer: Candido Martín González (Málaga)

Cited in 16 Documents

MSC:

17C05 Identities and free Jordan structures
17A15 Noncommutative Jordan algebras
17A30 Nonassociative algebras satisfying other identities
17A50 Free nonassociative algebras
17C50 Jordan structures associated with other structures
20C30 Representations of finite symmetric groups

Keywords:

quasi-Jordan algebras, associative dialgebras, computer algebra; polynomial identities; representations of the symmetric group

Citations:

Zbl 0845.16036; Zbl 0999.17002; Zbl 1188.17021; Zbl 1241.17001
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References:

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