FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 3, PAGES 57-78

**Zinbiel algebras under $q$-commutator**

A.
S.
Dzhumadil'daev

Abstract

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An algebra with the identity $t$_{1}(t_{2}t_{3}) =
(t_{1}t_{2} +
t_{2}t_{1})t_{3}
is called Zinbiel.
For example, $$**C**[x] under
multiplication $a\; \circ \; b\; =\; b\; \int $_{0}^{x} a dx is Zinbiel.
Let $a\; \circ $_{q} b =
a ∘ b + q b ∘ a be a $q$-commutator, where
$q$Î
**C**.
We prove that for any Zinbiel algebra $A$ the corresponding algebra
under commutator $A(-1)=\; (A,\; \circ $_{−1})
satisfies the identities $t$_{1}t_{2} =
-t_{2}t_{1}
and $(t$_{1}t_{2})(t_{3}t_{4}) +
(t_{1}t_{4})(t_{3}t_{2}) =
jac(t_{1},t_{2},t_{3})t_{4} +
jac(t_{1},t_{4},t_{3})t_{2},
where
$jac(t$_{1},t_{2},t_{3}) =
(t_{1}t_{2})t_{3} +
(t_{2}t_{3})t_{1} +
(t_{3}t_{1})t_{2}.
We find basic identities for $q$-Zinbiel algebras and
prove that they form varieties equivalent to the variety of Zinbiel
algebras if $q2$¹ 1.

Location: http://mech.math.msu.su/~fpm/eng/k05/k053/k05304h.htm

Last modified: September 14, 2005