Homology of depth-graded motivic Lie algebras and koszulity

TL;DR

This paper explores the homology of a motivic Lie algebra related to multizeta values, linking it to the Broadhurst-Kreimer conjecture and demonstrating equivalences with Koszulity and homology vanishing conditions.

Contribution

It establishes new equivalences between the homology conjecture, algebraic presentation, and Koszulity for a motivic Lie algebra, advancing understanding of its structure.

Findings

Part of the homology conjecture is equivalent to the algebra's presentation.

Remaining conjecture part is linked to the vanishing of a third homology group.

Proves that Koszulity relates to the homological properties of the algebra.

Abstract

The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.