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(t₂t₃) = (t₁t₂ + t₂t₁)t₃ 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 $t$₁t₂ = -t₂t₁ and $(t$₁t₂)(t₃t₄) + (t₁t₄)(t₃t₂) = jac(t₁,t₂,t₃)t₄ + jac(t₁,t₄,t₃)t₂, where $jac(t$₁,t₂,t₃) = (t₁t₂)t₃ + (t₂t₃)t₁ + (t₃t₁)t₂. 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.

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Last modified: September 14, 2005