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Vladimir Dotsenko

Publications and source records attributed to Vladimir Dotsenko.

At least 19 recordsLinked to original sources
arXiv2026

A Nielsen--Schreier variety of algebras without the PBW property

We prove the Nielsen--Schreier property for the variety of algebras defined by the identity $x(x^2)^2=(x^2)^2x$: every subalgebra of every free algebra in this variety is itself free. We also show that this variety of algebras does not have the Poincar\'e--Birkhoff--Witt property for universal multiplicative enveloping algebras. Our strategy of proof actually leads to infinitely many new varieties of non-associative algebras with the same behaviour. This offers new evidence supporting a conjecture of the first author and Umirbaev suggesting that the Nielsen--Schreier property over a field of zero characteristic is equivalent to freeness of universal multiplicative enveloping algebras of free algebras.

math.RA↗
arXiv2025

Algebraic versions of $\mathbb{T}^2$ and of $\mathbb{P}^1\times\mathbb{P}^1$ and Hochschild cohomology

We examine the Hochschild cohomology for triangular algebras that capture some aspects of geometry and topology of the torus and of the quadric surface, and for deformations of these algebras. In particular, this shows that the cup product on the Hochschild cohomology of a triangular algebra does not generally follow the intuition coming from monomial algebras. Our examples also demonstrate that the Hochschild cohomology of a deformation of an algebra may not experience the dimension drop but still have a different cup product structure, and that the Hochschild cohomologies of deformations of two derived equivalent algebras may exhibit noticeably different behaviours.

math.RA↗
arXiv2025

Derived operations satisfy standard identities

A derived operation is a bilinear operation on a commutative associative algebra $A$ defined intrinsically out of its product and several derivations of the product. We show that operators of left (or right) multiplications of a derived operation always satisfy a "standard identity" of certain order. In particular, it implies that each Rankin-Cohen bracket of modular forms, as well as each higher bracket of Kontsevich's universal deformation quantization formula for Poisson structures on $\mathbb{R}^n$, satisfies standard identities.

math.RA↗
arXiv2025

Yamaguti algebras and noncrossing partitions

Recently, Das defined a new type of algebras, the Yamaguti algebras, which are supposed to serve as envelopes of Lie-Yamaguti algebras appearing naturally in differential geometry. We show that the nonsymmetric operad of Yamaguti algebras admit a simple combinatorial description via noncrossing partitions without singleton blocks.

math.AT↗
arXiv2025

Operads of moduli spaces of points in $\mathbb{C}^d$ revisited

We study the operad structure on the homology of moduli spaces of pointed rooted trees of $d$-dimensional projective spaces, introduced by Chen, Gibney and Krashen a couple of decades ago. We describe this operad by generators and relations, show that it is homotopy Koszul, exhibit a Givental-type action on representations of that operad, and prove that this operad represents the homotopy quotient of the operad of chains of $S^1$-framed little $2d$-disks by its natural circle action. Our approach also sheds new light on the $d=1$ case, revealing a new combinatorial way to write the original Givental formulas.

math.AT↗
arXiv2025

On the conjecture of Kashuba and Mathieu about free Jordan algebras

Kashuba and Mathieu proposed a conjecture on vanishing of Lie algebra homology, implying a description of the $GL_d$-module structure of the free $d$-generated Jordan algebra. Their conjecture relies on a functorial version of the Tits--Kantor--Koecher construction that builds Lie algebras out of Jordan algebras. In this note, we summarize new intricate computational data concerning free Jordan algebras and explain why, despite a lot of overwhelmingly positive evidence, the conjecture of Kashuba and Mathieu is not true.

math.RA↗
arXiv2025

On the conjecture of Shang about free alternative algebras

Kashuba and Mathieu proposed a conjecture on vanishing of some components of the homology of certain Lie algebras, implying a description of the $GL_d$-module structure of the free $d$-generated Jordan algebra. Their conjecture relies on a functorial version of the Tits-Kantor-Koecher construction that builds Lie algebras out of Jordan algebras. Recently, Shang used a functorial construction of Allison, Benkart and Gao that builds Lie algebras out of alternative algebras to propose another conjecture on vanishing of some components of the homology of certain Lie algebras, implying a description of the $GL_d$-module structure of the free $d$-generated alternative algebra. In this note, we explain why the conjecture of Shang is not true.

math.RA↗
arXiv2024

Global Weyl modules for thin Lie algebras are finite-dimensional

The notion of Weyl modules, both local and global, goes back to Chari and Pressley in the case of affine Lie algebras, and has been extensively studied for various Lie algebras graded by root systems. We extend that definition to a certain class of Lie algebras graded by weight lattices and prove that if such a Lie algebra satisfies a natural "thinness" condition, then already the global Weyl modules are finite-dimensional. Our motivating example of a thin Lie algebra is the Lie algebra of polynomial Hamiltonian vector fields on the plane vanishing at the origin. We also introduce stratifications of categories of modules over such Lie algebras and identify the corresponding strata categories.

math.RT↗
arXiv2024

Volume preservation of Butcher series methods from the operad viewpoint

We study a coloured operad involving rooted trees and directed cycles of rooted trees that generalizes the operad of rooted trees of Chapoton and Livernet. We describe all the relations between the generators of a certain suboperad of that operad, and compute the Chevalley-Eilenberg homology of two naturally arising differential graded Lie algebras. This allows us to give short and conceptual new proofs of two important results on Butcher series methods of numerical solution of ODEs: absence of volume-preserving integration schemes and the acyclicity of the aromatic bicomplex, the key step in a complete classification of volume-preserving integration schemes using the so called aromatic Butcher series.

math.CT↗
arXiv2024

Hidden structures behind ambient symmetries of the Maurer-Cartan equation

For every differential graded Lie algebra $\mathfrak{g}$ one can define two different group actions on the Maurer-Cartan elements: the ubiquitous gauge action and the action of $\mathrm{Lie}_\infty$-isotopies of $\mathfrak{g}$, which we call the ambient action. In this note, we explain how the assertion of gauge triviality of a homologically trivial ambient action relates to the calculus of dendriform, Zinbiel, and Rota-Baxter algebras, and to Eulerian idempotents. In particular, we exhibit new relationships between these algebraic structures and the operad of rational functions defined by Loday.

math.QA↗
arXiv2024

Distributive lattices of varieties of Novikov algebras

We prove that a variety of Novikov algebras has a distributive lattice of subvarieties if and only if the lattice of its subvarieties defined by identities of degree three is distributive, thus answering, in the case of Novikov algebras, a question of Bokut from about fifty years ago. As a byproduct, we classify all Koszul operads with one binary generator of which the Novikov operad is a quotient.

math.RA↗
arXiv2024

Chain rule symmetry for singular SPDEs

We characterise the chain rule symmetry for the geometric stochastic heat equations in the full subcritical regime for Gaussian and non-Gaussian noises. We show that the renormalised counter-terms that give a solution invariant under changes of coordinates are generated by iterations of covariant derivatives. The result was known only for space-time white noises, with a very specific proof that so far could not be extended to the general case. The key idea of the present paper is to change the perspective on several levels and to use ideas coming from operad theory and homological algebra. Concretely, we introduce the operad of Christoffel trees that captures the counter-terms of the renormalised equation; our main new insight is to describe the space of invariant terms homologically, using a suitable perturbation of the differential of the operadic twisting of that operad. As a consequence, we obtain the correct renormalisation for the quasi-linear KPZ equation in the subcritical regime completing the programme started by Hairer and Gerencser. Previously, the main algebraic tool used in the study of singular SPDEs were Hopf algebras of decorated trees; our work shows that operad theory and homological algebra add new powerful tools with immediate applications to open problems that were out of reach by other methods.

math.PR↗
arXiv2024

Categorification of quiver diagonalization and Koszul algebras

In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.

math.RT↗
arXiv2023

Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads

We prove a theorem that computes, for any augmented operad $\mathcal{O}$, the stable homology of the Lie algebra of derivations of the free algebra $\mathcal{O}(V)$ with twisted bivariant coefficients (here stabilization occurs as $\dim(V)\to\infty$) out of the homology of the wheeled bar construction of $\mathcal{O}$; this can further be used to prove uniform mixed representation stability for the homology of the positive part of that Lie algebra with constant coefficients. This result generalizes both the Loday-Quillen-Tsygan theorem on the homology of the Lie algebra of infinite matrices and the Fuchs stability theorem for the homology of the Lie algebra of vector fields. We also prove analogous theorems for the Lie algebras of derivations with constant and zero divergence, in which case one has to consider the wheeled bar construction of the wheeled completion of $\mathcal{O}$. Similarly to how cyclic homology of an algebra $A$ may be viewed as an additive version of the algebraic $K$-theory of $A$, our results hint at the additive $K$-theoretic nature of the wheeled bar construction.

math.AT↗
arXiv2023

Novikov algebras and multi-indices in regularity structures

In this work, we introduce multi-Novikov algebras, a generalisation of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras.

math.RA↗
arXiv2023

The three graces in the Tits--Kantor--Koecher category

A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of $\mathfrak{sl}_2$-modules that are sums of copies of the trivial and the adjoint representation. That category is not symmetric monoidal, and so one cannot apply the wealth of results available for algebras over operads. Motivated by a recent conjecture of the second author and Mathieu, we embark on the exploration of the extent to which that category ``pretends'' to be symmetric monoidal. To that end, we examine various homological properties of free associative algebras and free associative commutative algebras, and study the Lie subalgebra generated by the generators of the free associative algebra.

math.KT↗
arXiv2023

Identities for deformation quantizations of almost Poisson algebras

We propose an algebraic viewpoint of the problem of deformation quantization of the so called almost Poisson algebras, which are algebras with a commutative associative product and an antisymmetric bracket which is a bi-derivation but does not necessarily satisfy the Jacobi identity. From that viewpoint, the main result of the paper asserts that, by contrast with Poisson algebras, the only reasonable category of algebras in which almost Poisson algebras can be quantized is isomorphic to the category of almost Poisson algebras itself, and the trivial two-term quantization formula already gives a solution to the quantization problem.

math.QA↗