# q-Leibniz Algebras

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

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## Abstract

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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.

## How to cite

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

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.},
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
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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