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Anton Khoroshkin

Publications and source records attributed to Anton Khoroshkin.

36 records · Page 2Linked to original sources
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↗
arXiv2019

Peter-Weyl, Howe and Schur-Weyl theorems for current groups

The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems is provided by the (current version of) Howe duality.

math.RT↗
arXiv2016

Differentials on graph complexes II - hairy graphs

We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes by a mechanism which we call the waterfall mechanism. By this mechanism we can construct many previously unknown classes and provide a first glimpse at the tentative global structure of the hairy graph cohomology.

math.QA↗
arXiv2016

On quantizable odd Lie bialgebras

Motivated by the universal obstruction to the deformation quantization of Poisson structures in infinite dimensions we introduce the notion of quantizable odd Lie bialgebra. The main result of the paper is a construction of a highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so called quantizable Poisson structures.

math.QA↗
arXiv2015

Highest weight categories and Macdonald polynomials

The aim of this paper is to introduce the categorical setup which helps us to relate the theory of Macdonald polynomials and the theory of Weyl modules for current Lie algebras discovered by V.\,Chari and collaborators. We identify Macdonald pairing with the homological pairing on the ring of characters of the Lie algebra of currents $\mathbf{g}\otimes\mathbb{C}[x,ξ]$. We use this description in order to define complexes of modules whose Euler characteristic of characters coincide with Macdonald polynomials. We generalize this result to the case of graded Lie algebras with anti-involution. We show that whenever the BGG reciprocity holds for the corresponding category of modules then these complexes collapse to the modules concentrated in homological degree $0$. The latter modules generalizes the notion of Weyl modules for current Lie algebras and the notion of Verma modules in the BGG category $\mathcal{O}$. We give different criterions of BGG reciprocity and apply them to the Lie algebra of currents $\mathbf{g}\otimes\mathbb{C}[x]$ with $\mathbf{g}$ semisimple.

math.RT↗
arXiv2015

Differentials on graph complexes

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.

math.QA↗
arXiv2014

On generating series of finitely presented operads

Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

math.QA↗
arXiv2014

Characteristic classes of flags of foliations and Lie algebra cohomology

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an $n$-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes characteristic classes of flags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences we avoid the need to use the methods of FFG, and moreover we are able to describe all the symmetric powers at once.

math.KT↗
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.

math.KT↗
arXiv2012

Hypercommutative operad as a homotopy quotient of BV

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/$Δ$ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

math.QA↗
arXiv2012

Macdonald Polynomials and BGG reciprocity for current algebras

We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category $\cal O$ of modules for $\lie g$ and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of $\lie{sl}_{n+1}$.

math.RT↗
arXiv2012

Shuffle algebras, homology, and consecutive pattern avoidance

Shuffle algebras are monoids for an unconvential monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations.

math.CO↗
arXiv2011

Free resolutions via Gröbner bases

For associative algebras in many different categories, it is possible to develop the machinery of Gröbner bases. A Gröbner basis of defining relations for an algebra of such a category provides a "monomial replacement" of this algebra. The main goal of this article is to demonstrate how this machinery can be used for the purposes of homological algebra. More precisely, our approach goes in three steps. First, we define a combinatorial resolution for the monomial replacement of an object. Second, we extract from those resolutions explicit representatives for homological classes. Finally, we explain how to "deform" the differential to handle the general case. For associative algebras, we recover a well known construction due to Anick. The other case we discuss in detail is that of operads, where we discover resolutions that haven't been known previously. We present various applications, including a proofs 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.

math.KT↗
arXiv2010

Using homological duality in consecutive pattern avoidance

Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.

math.CO↗
arXiv2010

Anick-type resolutions and consecutive pattern avoidance

For permutations avoiding consecutive patterns from a given set, we present a combinatorial formula for the multiplicative inverse of the corresponding exponential generating function. The formula comes from homological algebra considerations in the same sense as the corresponding inversion formula for avoiding word patterns comes from the well known Anick's resolution.

math.CO↗
arXiv2009

Gröbner bases for operads

We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads, and present operadic versions of Bergman's Diamond Lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads.

math.QA↗
arXiv2007

Quadratic algebras related to the bihamiltonian operad

We prove the conjectures on dimensions and characters of some quadratic algebras stated by B$.$L$.$Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad.

math.RA↗