Zinbiel algebras under q-commutators

Abstract

An algebra with the identity t1(t2t3) = (t1t2+t2t1)t3 is called Zinbiel. For example, ℂ[x] under the multiplication \(a\bigcirc b = b\int\limits_0^x {adx} \) is Zinbiel. Let a ○qb = a ○ b + q b ○ a be a q-commutator, where q ∈ ℂ. We prove that for any Zinbiel algebra A the corresponding algebra under the commutator A(−1) = (A, ○−1) satisfies the identities t1t2 = −t2t1 and (t1t2)(t3t4) + (t1t4)(t3t2) = jac(t1, t2, t3)t4 + jac(t1, t4, t3)t2, where jac(t1, 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.