Related papers: Fine structures inside the PreLie operad revisited
We establish, by elementary means, the existence of a cofibrantly generated monoidal model structure on the category of operads. By slicing over a suitable operad the classical Rezk model structure on the category of small categories is…
In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the…
The associative operad is a certain algebraic structure on the sequence of group algebras of the symmetric groups. The weak order is a partial order on the symmetric group. There is a natural linear basis of each symmetric group algebra,…
A pre-Lie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the pre-Lie product. We show that it…
We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the…
We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad of the endomorphism operad of $X$. We generalise this…
We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the…
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and…
We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. We apply it to several new monoidal categories that appeared recently in the theory of…
Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure…
We construct generalized multicategories associated to an arbitrary operad in Cat that is $\Sigma$-free. The construction generalizes the passage to symmetric multicategories from permutative categories, which is the case when the operad is…
Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be triangular coboundary. This…
Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras…
We prove that every automorphism of the category of free Lie algebras is a semi-inner automorphism.
We discuss natural operations on loops in a quasi-surface and show that these operations define a structure of a quasi-Lie bialgebra in the module generated by the set of free homotopy classes of non-contractible loops.
We develop Lie theory in the category $\text{Ver}_4^+$ over a field of characteristic 2, the simplest tensor category which is not Frobenius exact, as a continuation of arXiv:2406.10201. We provide a conceptual proof that an operadic Lie…
We will introduce the notion of higher derived bracket construction in the category of operads and prove that the higher derived bracket construction of Lie operad is equivalent to the cobar construction of Leibniz operad. The theorem is…
We describe automorphisms, derivations, and Rota -- Baxter operators on the series of simple pre-Lie algebras found by D. Burde in 1998.
We prove that all definable pre-orders are atomic, in a finitely generated free algebra of a discriminator variety of finite similarity type which is generated by its finite members.
In this paper we prove a ``Leray theorem'' for preLie algebras. We define a notion of ''Hopf'' preLie algebra: it is a preLie algebra together with a non associative permutative coproduct D and a compatibility relation between the preLie…