ZINBIEL ALGEBRAS UNDER q-COMMUTATORS

摘要

An algebra with the identity t_1 (t_2t_3) = (t_1t_2 + t_2t_1)t_3 is called Zinbiel. For example, C[x] under the multiplication a o b = b ∫_0~x a dx is Zinbiel. Let ao~qb = a o b + q b o 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, o_(-1)) satisfies the identities t_1t_2 = —t_2t_1 and (t_1t_2)(t_3t_4) + (t_1t_4)(t_3t_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_1t_2)t_3 + (t_2t_3)t_1 + (t_3t_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 q~2 ≠ 1.