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PBW-Pairs of Varieties of Linear Algebras

The notion of a Poincaré–Birkhoff–Witt (PBW)-pair of varieties of linear algebras over a field is under consideration. Examples of PBW-pairs are given. We prove that if (𝒱, 𝒲) is a PBW-pair and the variety 𝒱 is homogeneous and Schreier, then so is 𝒲; the results similar to the Schreier property for PBW-pairs are also true for the Freiheitssatz and Word problem. In particular, it follows that the Freiheitssatz is true for the varieties of Akivis and Sabinin algebras. We give also examples of varieties that do not satisfy the Freiheitssatz. It is shown that an element u of a free algebra 𝒲[X] in a homogeneous Schreier variety of algebras 𝒲 satisfying the Freiheitssatz is a primitive element (a coordinate polynomial) if and only if the factor algebra of 𝒲[X] by the ideal generated by the element u is a free algebra in 𝒲. We consider also properties of primitive elements.