Gröbner–Shirshov bases for Lie Ω-algebras and free Rota–Baxter Lie algebras

Abstract

We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov [Formula: see text]-words on a set [Formula: see text] and prove that the set of all the nonassociative Lyndon–Shirshov [Formula: see text]-words forms a linear basis of the free Lie [Formula: see text]-algebra on the set [Formula: see text]. From this, we establish Gröbner–Shirshov bases theory for Lie [Formula: see text]-algebras. As applications, we give Gröbner–Shirshov bases of a free [Formula: see text]-Rota–Baxter Lie algebra, of a free modified [Formula: see text]-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebra and, then linear bases of these three algebras are obtained.