arXiv subjects

Anton Khoroshkin

Publications and source records attributed to Anton Khoroshkin.

At least 19 recordsLinked to original sources

arXiv2026

On finite dimensionality of homology of subalgebras of vector fields

We show that finite tensor products of modules of tensor fields are Noetherian modules over any graded Lie subalgebra of finite codimension in the Lie algebra of polynomial vector fields on $\mathbb{R}^n$. As a corollary, we prove the conjecture of I.\,M. Gelfand, announced at the ICM in Nice in 1970, on the finite-dimensionality of the continuous cohomology of graded Lie subalgebras of finite codimension in the Lie algebra of formal vector fields $W_n$.

math.QA↗

arXiv2026

Bubble sort and Howe duality for staircase matrices

We prove the alternating Cauchy identity for staircase matrices conjectured in arXiv:2411.03117, together with an explicit description of the coefficients occurring in it. As a byproduct, our approach also yields a new, independent (more combinatorial) proof of the Cauchy identities for staircase matrices established in arXiv:2411.03117. The first part of the paper focuses on combinatorial aspects. It is self-contained, of independent interest, and introduces a generalization of parabolic Bruhat graphs for monotone functions on an arborescent poset. The second part centers on representation theory. We propose a generalization of the classical Howe duality for staircase matrices in terms of the distributive lattice of Demazure submodules within a given integrable representation. Computing the associated character yields all desired Cauchy identities for staircase matrices.

math.RT↗

arXiv2026

Cauchy identities for staircase matrices

The well known Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $λ$ of products of Schur polynomials: $s_λ(x)s_λ(y)$. Algebraically, this identity comes from the decomposition of the symmetric algebra of the space of rectangular matrices, considered as a $\mathfrak{gl}_m$-$\mathfrak{gl}_n$-bimodule. We generalize the Cauchy decomposition by replacing rectangular matrices with arbitrary staircase-shaped matrices equipped with the left and right actions of the Borel upper-triangular subalgebras. For any given staircase shape $\mathsf{Y}$ we describe left and right ``standard" filtrations on the symmetric algebra of the space of shape $\mathsf{Y}$ matrices. We show that the subquotients of these filtrations are tensor products of Demazure and opposite van der Kallen modules over the Borel subalgebras. On the level of characters, we derive two distinct expansions for the product $(1 - x_i y_j)^{-1}$ for $(i,j) \in \mathsf{Y}$ written as sums of products of key polynomials $κ_λ(x)$ and (opposite) Demazure atoms $a^μ(y)$.

math.RT↗

arXiv2026

Input/output coloring and Gröbner basis for dioperads

We introduce a functor $Ψ$ that associates to a dioperad $P$ acting on a vector space $V$ a two-colored operad $Ψ(P)$ acting on the pair $(V, V^*)$. The construction is based on a simple pictorial idea: by selecting one input or output and dualizing, if necessary, the remaining ones, any dioperadic tree can be ``rerooted'' as a colored operadic tree. This transformation allows one to apply the standard operadic machinery -- such as Gröbner bases and Hilbert series -- to the study of dioperads. We illustrate the method with several examples and applications. (1) We compute the dimensions of the spaces of operations for the dioperad of Lie bialgebras. (2) We describe a Gröbner basis and construct a minimal resolution for the dioperad of triangular Lie bialgebras. (3) We perform explicit computations for the dioperad of ``algebraic string operations''. (4) We give a pictorial construction proving the existence of quadratic Gröbner bases and establishing the Koszul property for a broad class of dioperads arising from cyclic operads.

math.QA↗

arXiv2026

Graphical configuration spaces, Contractads and Formality

Given a finite simple connected graph $Γ$, the graphical configuration space $\mathrm{Conf}_Γ(X)$ is the space of collections of points in $X$ indexed by the vertices of $Γ$, where points corresponding to adjacent vertices must be distinct. When $X=\mathbb{R}^d$ and the points are replaced by small disks, the resulting spaces for all possible graphs fit together into an algebraic structure that extends the little disks operad, called the little disks contractad $\mathcal{D}_d$. In this paper, we investigate the homotopical and algebraic properties of the little disks contractad $\mathcal{D}_d$. We construct and study Fulton-MacPherson compactifications of graphical configuration spaces, which provide a convenient model for $\mathcal{D}_d$ within the class of compact manifolds with boundary. Using these and wonderful compactifications, we prove that $\mathcal{D}_d$ is formal in the category of (Hopf) contractads for $d=1$, $d=2$, and for chordal graphs for any $d$. We also identify the first obstructions to coformality in the case of cyclic graphs. In addition, we give a combinatorial description of the cell structure of $\mathcal{D}_2$ and present applications to the study of graphical configuration spaces $\mathrm{Conf}_Γ(X)$ using the language of twisted algebras.

math.AT↗

arXiv2026

The properad of quadratic Poisson structures is Koszul

In this paper, we suggest a sufficient condition on the properadic envelope of a quadratic dioperad to be Koszul in terms of twisted associative algebras. As a particular new example, we show that the properad of quadratic Poisson structures is Koszul.

math.QA↗

arXiv2025

Real models for the framed little $n$-disks operads

We study the action of the orthogonal group on the little $n$-disks operads. As an application we provide small models (over the reals) for the framed little $n$-disks operads. It follows in particular that the framed little $n$-disks operads are formal (over the reals) for $n$ even and coformal for all $n$.

math.QA↗

arXiv2025

Peter-Weyl theorem for Iwahori groups and highest weight categories

We study the algebra of functions on the Iwahori group via the category of graded bounded representations of its Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized Weyl modules. Using this identification we express the characters of the standard and costandard objects in terms of specialized nonsymmetric Macdonald polynomials. We also prove that our category of interest admits a generalized highest weight structure (known as stratified structure). We show, more generally, that such a structure on a category of representations of a Lie algebra implies the Peter-Weyl type theorem for the corresponding algebraic group. In the Iwahori case, standard filtrations of indecomposable projective objects correspond to new ``reciprocal'' Macdonald-type identities.

math.RT↗

arXiv2024

Real moduli space of stable rational curves revised

The real locus of the moduli space of stable genus-zero curves with marked points, $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$, is known to be a smooth manifold and is the Eilenberg-MacLane spaces for the so-called pure Cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we show that the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal. As an application of these operadic constructions, we prove that for each $n$, the cohomology ring $H^{\bullet}(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}), {\mathbb{Q}})$ is a Koszul algebra, and that the manifold $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal for $n\geq 6$ but is a rational $K(π,1)$-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure Cactus groups.

math.AT↗

arXiv2024

Categorification of DAHA and Macdonald polynomials

We describe a categorification of the Double Affine Hecke Algebra (${\mathcal{H}\kern -.4em\mathcal{H}}$) associated with an affine Lie algebra $\widehat{\mathfrak{g}}$, including a categorification of the polynomial representation and Macdonald polynomials. Our categorification results are presented in the derived setting, focusing on the derived category of graded modules over the Lie superalgebra ${\mathfrak I}[ξ]$, where ${\mathfrak I} \subset \widehat{\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $ξ$ is a formal odd variable. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${\mathfrak{p}}_{i}$ categorify the Demazure operators $T_i + 1 \in {\mathcal{H}\kern -.4em\mathcal{H}}$, ensuring that all algebraic relations of $T_i$ have categorical interpretations. Second, for each dominant weight $λ$ we introduce a complex ${\mathbb{EM}}_λ$ of ${\mathfrak{I}}[ξ]$-modules and a complex ${\mathbb{PM}}_λ$ of ${\mathfrak{g}}[z,ξ]$-modules, whose Euler characteristics are equal to nonsymmetric $E_λ$ and symmetric $P_λ$ Macdonald polynomials respectively. We illustrate our theory with the example $\mathfrak{g}=\mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${\mathfrak{I}}[ξ]$ such that their supercharacters coincide with certain normalizations of nonsymmetric Macdonald polynomials.

math.RT↗

arXiv2024

Hilbert series for contractads and modular compactifications

Contractads are operadic-type algebraic structures well-suited for describing configuration spaces indexed by a simple connected graph $Γ$. Specifically, these configuration spaces are defined as $\mathrm{Conf}_Γ(X):=X^{|V(Γ)|}\setminus \cup_{(ij)\in E(Γ)} \{x_i=x_j\}$. In this paper, we explore functional equations for the Hilbert series of Koszul dual contractads and provide explicit Hilbert series for fundamental contractads such as the commutative, Lie, associative and the little discs contractads. Additionally, we focus on a particular contractad derived from the wonderful compactifications of $\mathrm{Conf}_Γ(\mathbb{k})$, for $\mathbb{k}=\mathbb{R},\mathbb{C}$. First, we demonstrate that for complete multipartite graphs, the associated wonderful compactifications coincide with the modular compactifications introduced by Smyth. Second, we establish that the homology of the complex points and the homology of the real locus of the wonderful contractad are both quadratic and Koszul contractads. We offer a detailed description of generators and relations, extending the concepts of the Hypercommutative operad and cacti operads, respectively. Furthermore, using the functional equations for the Hilbert series, we describe the corresponding Hilbert series for the homology of modular compactifications.

math.QA↗

arXiv2023

Parahoric Lie algebras and parasymmetric Macdonald polynomials

The main goal of this paper is to categorify the specialized parasymmetric (intermediate) Macdonald polynomials. These polynomials depend on a parabolic subalgebra of a simple Lie algebra and generalize the symmetric and nonsymmetric Macdonald polynomials. To achieve this we introduce cyclic modules of the parahoric subalgebras of the affine Kac-Moody Lie algebras such that their characters coincide with the specializations of the parasymmetric polynomials at zero and infinity. These cyclic modules are proved to coincide with standard and costandard objects in certain categories of representations of parahoric algebras. We show that the categories in question are stratified, i.e. they are graded highest weight categories. As a consequence, we derive an analog of the Peter-Weyl theorem describing the bi-module of functions on the parahoric and parabolic Lie groups via the mentioned above standard and costandard modules.

math.RT↗

arXiv2022

On deformation quantization of quadratic Poisson structures

We study the deformation complex of the dg wheeled properad of $\mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of $\mathbb{Z}$-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps. In particular we show that two universal quantizations of Poisson structures are equivalent if the agree on generic quadratic Poisson structures.

math.QA↗

arXiv2021

Homotopical rigidity of the pre-Lie operad

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005, including a Lie-theoretic version of the Deligne conjecture.

math.KT↗

arXiv2020

Gröbner Bases for Coloured Operads

In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Gröbner bases for coloured operads. Among the examples for which we show the existance of a quadratic Gröbner basis we consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).

math.CT↗

arXiv2020

Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)

We define derived Poincaré--Birkhoff--Witt maps of dg operads or derived PBW maps, for short, which extend the definition of PBW maps between operads of V.~Dotsenko and the second author in 1804.06485, with the purpose of studying the universal enveloping algebra of dg Lie algebras as a functor on the homotopy category. Our main result shows that the map from the homotopy Lie operad to the homotopy associative operad is derived PBW, which gives us an amenable description of the homology of the universal envelope of an $L_\infty$-algebra in the sense of Lada--Markl. We deduce from this several known results involving universal envelopes of $L_\infty$-algebras of V. Baranovsky and J. Moreno-Fernández, and extend D. Quillen's classical quasi-isomorphism $\mathcal C \longrightarrow BU$ from dg Lie algebras to $L_\infty$-algebras; this confirms a conjecture of J. Moreno-Fernández.

math.KT↗

arXiv2020

PBW property for associative universal enveloping algebras over an operad

Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $\mathcal{P}$ is discovered. Moreover, given any symmetric operad $\mathcal{P}$, together with a Gröbner basis $G$, a condition is given in terms of the structure of the underlying trees associated with leading monomials of $G$, sufficient for the PBW property to hold. Examples are provided.

math.QA↗

arXiv2019

Quadratic Algebras arising from Hopf operads generated by a single element

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.

math.QA↗