Higman-Neumann-Neumann extension and embedding theorems for Leibniz algebras

In this work we introduce the Higman-Neumann-Neumann (HNN)- extensions and the appropriate embedding theorems for dialgebras and Leibniz algebras. Due to the importance of the connection between the dialgebras and Leibniz algebras and the relationship between associative algebras and Lie algebras, we recall the theory of Groebner-Shirshov bases, and the Composition-Diamond Lemma in associative algebras and Lie algebras, as well as the theory of Groebner-Shirshov bases for dialgebras. As an application of the HNN-extensions of dialgebras and Leibniz algebras, we provide embedding theorems for dialgebras and Leibniz algebras, respectively: every dialgebra embeds inside its any HNN-extension and every Leibniz algebra embeds inside its any HNN-extension

A Gröbner-Basis Theory for Divide-and-Conquer Recurrences

International audienceWe introduce a variety of noncommutative polynomials that represent divide-and-conquer recurrence systems. Our setting involves at the same time variables that behave like words in purely noncom-mutative algebras and variables governed by commutation rules like in skew polynomial rings. We then develop a Gröbner-basis theory for left ideals of such polynomials. Strikingly, the nature of commutations generally prevents the leading monomial of a polynomial product to be the product of the leading monomials. To overcome the difficulty, we consider a specific monomial ordering, together with a restriction to monic divisors in intermediate steps. After obtaining an analogue of Buchberger's algorithm, we develop a variant of the 4 algorithm, whose speed we compare

Associator dependent algebras and Koszul duality

We resolve a ten year old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras.Comment: 20 pages, submitted versio

Pushforwards via scattering equations with applications to positive geometries

© 2022 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), https://creativecommons.org/licenses/by/4.0/In this paper we explore and expand the connection between two modern descriptions of scattering amplitudes, the CHY formalism and the framework of positive geometries, facilitated by the scattering equations. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry defined in the CHY moduli space. In order to compute these canonical forms in kinematic spaces, we study the general problem of pushing forward arbitrary rational differential forms via the scattering equations. We develop three methods which achieve this without ever needing to explicitly solve any scattering equations. Our results use techniques from computational algebraic geometry, including companion matrices and the global duality of residues, and they extend the application of similar results for rational functions to rational differential forms.Peer reviewe

Combinatorics of pre-Lie products sharing a Lie bracket

We study in detail the operad controlling several pre-Lie algebra structures sharing the same Lie bracket. Specifically, we show that this operad admits a combinatorial description similar to that of Chapoton and Livernet for the pre-Lie operad, and that it has many of the remarkable algebraic properties of the pre-Lie operad.Comment: 40 pages, 9 figure