Geodesic
q-Leibniz Algebras
Serdica Mathematical Journal, Tome 34 (2008) no. 2, pp. 415-440.

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An algebra (A,ο) is called Leibniz if aο(bοc) = (a ο b)ο c-(a ο c) ο b for all a,b,c ∈ A. We study identities for the algebras A(q) = (A,οq), where a οq b = a ο b+q b ο a is the q-commutator. Let Char K ≠ 2,3. We show that the class of q-Leibniz algebras is defined by one identity of degree 3 if q2 ≠ 1, q ≠−2, by two identities of degree 3 if q = −2, and by the commutativity identity and one identity of degree 4 if q = 1. In the case of q = −1 we construct two identities of degree 5 that form a base of identities of degree 5 for −1-Leibniz algebras. Any identity of degree 5 for −1-Leibniz algebras follows from the anti-commutativity identity.
Keywords: Leibniz Algebras, Zinbiel Algebras, Omni-Lie Algebras, Polynomial Identities, q-Commutators 
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     author = {Dzhumadil'daev, A. S.},
     title = {q-Leibniz {Algebras}},
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Dzhumadil'daev, A. S. q-Leibniz Algebras. Serdica Mathematical Journal, Tome 34 (2008) no. 2, pp. 415-440. http://geodesic.mathdoc.fr/item/SMJ2_2008_34_2_a3/