Abstract
We first construct linear bases for free Lie-admissible algebras and develop a new theory of Gröbner–Shirshov bases for Lie-admissible algebras. Then we prove an analogue of the Poincaré–Birkhoff–Witt theorem, that is, every Lie algebra L can be embedded into its universal enveloping Lie-admissible algebra U(L), where the basis of U(L) does not depend on the multiplication table of L. Finally, we show that the basic rank of the variety of Lie-admissible algebras is 1. As a corollary, the universal enveloping Lie-admissible algebra of an abelian Lie algebra does not satisfy any nontrivial identity in the variety of Lie-admissible algebras.
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