# q-Leibniz Algebras

Serdica Mathematical Journal (2008)

- Volume: 34, Issue: 2, page 415-440
- ISSN: 1310-6600

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topDzhumadil'daev, A. S.. "q-Leibniz Algebras." Serdica Mathematical Journal 34.2 (2008): 415-440. <http://eudml.org/doc/281417>.

@article{Dzhumadildaev2008,

abstract = {2000 Mathematics Subject Classification: Primary 17A32, Secondary 17D25.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.},

author = {Dzhumadil'daev, A. S.},

journal = {Serdica Mathematical Journal},

keywords = {Leibniz Algebras; Zinbiel Algebras; Omni-Lie Algebras; Polynomial Identities; q-Commutators},

language = {eng},

number = {2},

pages = {415-440},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {q-Leibniz Algebras},

url = {http://eudml.org/doc/281417},

volume = {34},

year = {2008},

}

TY - JOUR

AU - Dzhumadil'daev, A. S.

TI - q-Leibniz Algebras

JO - Serdica Mathematical Journal

PY - 2008

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 34

IS - 2

SP - 415

EP - 440

AB - 2000 Mathematics Subject Classification: Primary 17A32, Secondary 17D25.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.

LA - eng

KW - Leibniz Algebras; Zinbiel Algebras; Omni-Lie Algebras; Polynomial Identities; q-Commutators

UR - http://eudml.org/doc/281417

ER -

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