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Vladimir Dotsenko

Publications and source records attributed to Vladimir Dotsenko.

At least 55 records · Page 3Linked to original sources
arXiv2019

Endofunctors and Poincaré-Birkhoff-Witt theorems

We determine what appears to be the bare-bones categorical framework for Poincaré-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a Poincaré-Birkhoff-Witt property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new Poincaré-Birkhoff-Witt type theorems. In particular, we prove a PBW type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.

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arXiv2019

Distributive laws between the operads Lie and Com

Using methods of computer algebra, especially Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet-Loday formula deforming the Poisson operad into the associative operad.

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arXiv2019

Finite generation for Hochschild cohomology of Gorenstein monomial algebras

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann-Holloway-Snashall-Solberg-Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

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arXiv2019

Non-Koszulness of operads and positivity of Poincaré series

We prove that the operad of mock partially associative $n$-ary algebras is not Koszul, as conjectured by the second and the third author in 2009, and utilise the Zeilberger's algorithm for hypergeometric summation to demonstrate that non-Koszulness of that operad cannot be established by hunting for negative coefficients in the inverse of its Poincaré series.

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arXiv2019

Word operads and admissible orderings

We use Giraudo's construction of combinatorial operads from monoids to offer a conceptual explanation of the origins of Hoffbeck's path sequences of shuffle trees, and use it to define new monomial orders of shuffle trees. One such order is utilised to exhibit a quadratic Gröbner basis of the Poisson operad.

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arXiv2019

Functorial PBW theorems for post-Lie algebras

Using the categorical approach to Poincaré-Birkhoff-Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota-Baxter algebras of tridendriform algebras, for universal enveloping Rota--Baxter Lie algebras of post-Lie algebras, and for universal enveloping tridendriform algebras of post-Lie algebras. Similar results, though without functoriality of the PBW isomorphisms, were recently obtained by Gubarev. Our methods are completely different and mainly rely on methods of rewriting theory for shuffle operads.

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arXiv2019

The twisting procedure

This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, we settle the integration theory of complete pre-Lie algebras in order to describe this twisting procedure in terms of gauge group action. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of (homotopy) algebras. We also give a new presentation of the twisting procedure for operads à la Willwacher and we perform new homology computations of graph complexes.

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arXiv2018

Toric varieties of Loday's associahedra and noncommutative cohomological field theories

We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, coming from the fact that they admit several equivalent interpretations: as the toric varieties of Loday's realisations of the associahedra, as the brick manifolds recently defined by Escobar, and as the De Concini-Procesi wonderful models for certain subspace arrangements.

math.AG↗
arXiv2017

Algebraic structures of $F$-manifolds via pre-Lie algebras

We relate the operad FMan controlling the algebraic structure on the tangent sheaf of an $F$-manifold (weak Frobenius manifold) defined by Hertling and Manin to the operad PreLie of pre-Lie algebras: for the filtration of PreLie by powers of the ideal generated by the Lie bracket, the associated graded object is FMan.

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arXiv2017

Veronese powers of operads and pure homotopy algebras

We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie $k$-algebras and Lie triple systems. In the case of Lie $k$-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg--Kapranov power series test is inconclusive.

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arXiv2017

Classification of regular parametrized one-relation operads

Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{σ\in S_3}x_σ\, a_{σ(1)}(a_{σ(2)}a_{σ(3)})\ ; \] such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_σ\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the identity $((a_1a_2)a_3)=0$, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

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arXiv2017

Boardman--Vogt tensor products of absolutely free operads

We establish a combinatorial model for the Boardman--Vogt tensor product of several absolutely free operads, that is free symmetric operads that are also free as $\mathbb{S}$-modules. Our results imply that such a tensor product is always a free $\mathbb{S}$-module, in contrast with the results of Kock and Bremner--Madariaga on hidden commutativity for the Boardman--Vogt tensor square of the operad of non-unital associative algebras.

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arXiv2016

A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula

We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the left adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilised to provide examples showing limitations of two standard tools of the Koszul duality theory. We also apply these functors to get a homotopical algebra proof of the Lagrange inversion formula.

math.CT↗
arXiv2016

Pre-Lie deformation theory

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the $dd^c$-lemma.

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arXiv2015

A tale of three homotopies

For a Koszul operad $\mathcal{P}$, there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy $\mathcal{P}$-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.

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arXiv2014

Givental Action and Trivialisation of Circle Action

In this paper, we show that the Givental group action on genus zero cohomological field theories, also known as formal Frobenius manifolds or hypercommutative algebras, naturally arises in the deformation theory of Batalin--Vilkovisky algebras. We prove that the Givental action is equal to an action of the trivialisations of the trivial circle action. This result relies on the equality of two Lie algebra actions coming from two apparently remote domains: geometry and homotopical algebra.

math.QA↗
arXiv2013

Quillen homology for operads via Gröbner bases

The main goal of this paper is to present a way to compute Quillen homology of operads. The key idea is to use the notion of a shuffle operad we introduced earlier; this allows to compute, for a symmetric operad, the homology classes and the shape of the differential in its minimal model, although does not give an insight on the symmetric groups action on the homology. Our approach goes in several steps. First, we regard our symmetric operad as a shuffle operad, which allows to compute its Gröbner basis. Next, we define a combinatorial resolution for the "monomial replacement" of each shuffle operad (provided by the Gröbner bases theory). Finally, we explain how to "deform" the differential to handle every operad with a Gröbner basis, and find explicit representatives of Quillen homology classes for a large class of operads. We also present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin-Vilkovisky algebras and of Rota-Baxter algebras.

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