Gröbner–Shirshov bases for brace algebras

ABSTRACT

Let A be a brace algebra. This structure implies that A is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. For each pre-Lie algebra L, we find a Gröbner–Shirshov basis for its universal brace algebra U_b(L). As applications, we determine an explicit linear basis for U_b(L) and prove that L is a pre-Lie subalgebra of U_b(L).

KEYWORDS:

• Brace algebra
• Gröbner–Shirshov basis
• pre-Lie algebra

2010 AMS MATHEMATICS SUBJECT CLASSIFICATION:

• 16S15
• 13P10
• 17A30

Acknowledgment

We are thankful to Leonid Bokut and Vladimir Dotsenko for useful remarks on a draft version of this paper and references to literature.