On Free Pre-Lie Algebras

In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal’s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative Gröbner-Shirshov basis S for a free pre-Lie algebra such that Irr(S) is a monomial basis (called normal words) of a free pre-Lie algebra, where Irr(S) is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.

Gröbner-Shirshov Bases for Plactic Algebras

A finite Gröbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Gröbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.