Anton Khoroshkin's papers, preprints, drafts and other.
Comments are very welcome.
Contact me by Email: akhoroshkin "at" scgp.stonybrook.edu or khoroshkin "at" gmail.com
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Please note that my point of view may contradict with the point of view of my coauthors.
Click on the description bottom and you will see my short overview of the corresponding paper.
By clicking on the hide bottom you will remove these comments from the screen.
Please note that my point of view may contradict with the point of view of my coauthors.
published, submitted or posted on arXiv:
Lie algebra cohomology representing characteristic classes of flags of foliations pdf, TeX, arXiv, description, hide
Submitted to Moscow Mathematical Journal
This article contains the computation of the Lie algebra cohomology of the infinite-dimensional Lie algebra of formal vector fields with coefficients in symmetric powers of the coadjoint representation. At the same time we compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The resulting cohomology are known to be responsible for the characteristic classes of the flags of foliations and are well used in the local Riemann-Roch theorem by Feigin and Tsygan and later on by Felder,Shoikhet and collaborators.
We use the degeneration theorems of appropriate Hochschild-Serre spectral sequences and provide the method which allows us to avoid one of the most complicated computation in the invariant theory which was done by Gelfand, Feigin and Fuchs in order to cover the case of first symmetric power. The method we use gives a uniform and beautiful answer for all symmetric powers at the same time.
We believe that the same method should provide a useful tool in the computation of the Lie algebra cohomology of the Lie algebra of vector fields vanishing at the origin and the Lie algebra cohomology of Lie superalgebra of currents $\mathbf{sl}_n\otimes \mathbb{C}[t,\xi]$. Where $t$ is an even variable and $\xi$ is odd.
MacDonald Polynomials and BGG reciprocity for current algebras : pdf, TeX, arXiv, description, hide
joint with Matthew Bennett, Arkady Berenstein, Vyjayanthi Chari, Sergey Loktev.
Submitted to Selecta Mathematica.
In this article we proved the BGG duality for Weyl modules for the current Lie algebra $sl_n\otimes\mathbb{C}[t]$.
We study the category of graded representations with finite-dimensional graded pieces for the current algebra associated to a simple Lie algebra. Denote by $V({\lambda})$ the irreducible finite-dimensional $\mathbf{g}$-module with the highest weight $\lambda$. The projective cover $P({\lambda})$ of this module in the aforementioned category is known to coincide with the induced module $Ind_{\mathbf{g}}^{\mathbf{g}\otimes\mathbb{C}[t]} V({\lambda})$. The maximal (resp. maximal finite-dimensional) module with a highest weight vector $v_{\mu}$ is called the global (resp. local) Weyl module and is denoted by $W({\mu})$ (resp. $W_{loc}({\mu})$). We state the BGG duality for current algebras in the following manner:
For every dominant weight $\lambda$ the projective cover $P({\lambda})$ admits a filtration by global Weyl modules. Moreover, we have the following reciprocity for multiplicities for all pairs of dominant weights $\lambda$ and $\mu$: $$ [P({\lambda}): W({\mu})] = [W_{loc}(\mu):V(\lambda)]. $$ Namely, the multiplicity of the global Weyl modules with the highest weight $\mu$ in the associated graded to projective cover $P(\lambda)$ coincides with the multiplicity of $V({\lambda})$ in the Jordan-Holder decomposition of the local Weyl module with the highest weight $\mu$.
For $\mathbf{g}=\mathbf{sl}_n$ it is known that the character of a local Weyl module is equal to specialization of MacDonald polynomial for $t=0$. The global Weyl module with highest weight $\mu$ is a tensor product of local Weyl module and a graded vector space whose graded character coincides with the scalar product of MacDonald polynomial $P({\mu})$ with itself (once again we take the substitution $t=0$). Therefore, the BGG duality for Weyl modules provides a representational theoretic meaning of the MacDonald constant term identity for $t=0$.
Unfortunately, so far the proof uses the identity itself and the knowledge of characters of Weyl modules in $\mathbf{sl}_n$ case. Right now I discovered a uniform proof (to be typed soon) of the BGG duality for Weyl modules in case of all simple Lie algebras $\mathbf{g}$ that provides both a character formula for Weyl modules and a representational theoretic meaning of the MacDonald constant term identity for $t=0$. The proof is based on categorical implications that are valid for all highest weight categories. In order to use the appropriate general formalism it is sufficient to prove vanishing of appropriate extensions groups.
We give an explicit formula for a quasi-isomorphism between the operads $\mathsf{Hycomm}$ (the homology of the moduli space of stable genus $0$ curves) and $\mathsf{BV}/\mathsf{Delta}$ (the homotopy quotient of Batalin-Vilkovisky operad by the $\mathsf{BV}$-operator). In other words we derive an equivalence of $\mathsf{Hycomm}$-algebras and $\mathsf{BV}$-algebras enhanced with a homotopy that trivializes the $\mathsf{BV}$-operator.
In particular, the desired isomorphism provides a possible way to define a complicated structure of a $\mathsf{Hycomm}$-algebra using a simpler structure of a $\mathsf{BV}$--algebra. Recall, that $\mathsf{Hycomm}$-operad has generators with arbitrary many inputs and $\mathsf{BV}$-operad has generating operations with arities $1$ and $2$. The price for this description is normally the infinitness of the dimension of the corresponding $\mathsf{BV}$-algebra.
The formulas for quasi-isomorphism 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 other way round. We go through a chain of explicit formulas on resolutions of $\mathsf{Hycomm}$ and $\mathsf{BV}$ and the Givental graphs appears by nature. In particular, we get a homological explanation of the Givental group action on $\mathsf{Hycomm}$-algebras.
The second approach seems to be a technical homological computation, but the initial idea is quite simple and I insist that this computation is very important in order to understand several things. First, the nature of isomorphism itself, second, the mystery of Givental formulas, third, the action of $\psi$-classes.
With any given operad $\mathcal{P}=\cup_{i=1}^{\infty}\mathcal{P}(n)$ we can associate a generating series of dimensions of the space of operations with the same arity. This article is an attempt to find a reasonable bounds and recursive relations for these generating series. Of course, for arbitrary operad the corresponding series may be transcendental, therefore we restrict our self to the case of operads that admits a finite Grobner basis. Recall, that there exists the theory of monomials and Grobner bases for nonsymmetric operads and there is no corresponding theory for symmetric operads. In order to avoid this problem one has to forget part of the action of symmetric group. The latter theory is called shuffle operads and is described in my joint paper.
The generating series of an operad with a chosen finite Grobner basis and a generating series of the associated graded operad with monomial relations are the same. Therefore, the problem we consider reduces to the description of generating series of monomial operads.
The main result of this note looks as follows:
The ordinary generating series $\sum_{n=1}^{\infty} dim{\mathcal{P}}(n) t^n$ of a finitely presented monomial nonsymmetric operad ${\mathcal{P}}$ is an algebraic function.
The exponential generating series $\sum_{n=1}^{\infty} dim{\mathcal{P}}(n) \frac{t^{n}}{n!}$ of a finitely presented monomial shuffle operad ${\mathcal{P}}$ is differential algebraic function if the set of relations is closed under shuffle-permutations. The shuffle-permutations are those permutations which permutes the labels of a planar tree-monomial, but do not change the underlying planar tree.
The proofs are constructive. Namely, we present algorithms on how to find a finite system of recursive equations. The positivity of the coefficients in these systems implies some restrictions on the generating series with small growths:
If (in addition to aforementioned finiteness assumptions) the growth of the dimensions $\mathcal{P}(n)$ 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 $\{ dim {\mathcal{P}}(n) \}$ is rational.
All algorithms are provided by series of examples. Being inspired by hunting examples out of the PBW case we have discovered several non-quadratic examples that have their own interest.
The main goal of this paper is to present a way to compute Quillen homology of a shuffle operad with a known Grobner basis. Similar to the strategy taken in a celebrated paper of David Anick, our approach goes in several steps. We define a combinatorial resolution for the ``monomial replacement'' of a shuffle operad, explain how to ``deform'' the differential to handle the general case, and find explicit representatives of Quillen homology for a large class of operads with monomial relations.
We 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.
The method of writing a resolution presented in this paper is very general. Namely, whenever you have a category which admits a theory of monomials (including Grobner bases and Buchberger algorithm) you can do the same procedure:
First, take an object with a chosen Grobner basis. Second, define a resolution for an object with monomial relations (the monomial replacement of the starting object). Third, lower terms of relations will affect additional summands in the description of the differential in the resolution.
We tried to handle a general case here.
Abstract: For associative algebras in many different categories, it is possible to develop the machinery of Grobner bases. A Grobner 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.
This paper as well as two subsequent preprints deals with consecutive pattern avoidance.
Recall that, in combinatorics, one suggests to consider the following notion of divisibility for permutations. We say that the permutation $\sigma\in S_n$ is divisible by $\pi\in S_m$ iff there exists an interval $[i+1,i+m]$ of length $m$ such that the integers in the sequence $(\sigma(i),\ldots,\sigma(i+m-1))$ has the same local order as the integers in the sequence $(\pi(1),\ldots,\pi(m))$. In other words $\pi(s)>\pi(t) \Leftrightarrow \sigma(i+s)>\sigma(i+t)$. We say that $\sigma$ is divisible from the left (resp. from the right) by $\pi$ iff in the above assumptions the interval $[i+1,i+m]$ is the leftmost (resp. rightmost) sub-interval of $[1,n]$ of length $m$. Finally, we say that $\sigma$ avoids $\pi$ iff $\sigma$ is not divisible by $\pi$.
We proved that the generating series for the number of permutations avoiding elements of a given collection of patterns from a given set $S$ depends only on the combinatorics of the overlappings of patterns in these sets. We say that $\nu$ is an overlapping of $\pi_1$ and $\pi_2$ if $\nu$ is divisible from the left by $\pi_1$ and divisible from the right by $\pi_2$ but the length of $\nu$ is strictly less than the sum of lengths of $\pi_1$ and $\pi_2$.
We present 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.
This paper has been started as a particular application of the method of resolutions via Grobner bases we suggested here.
We introduce a notion of a shuffle algebra. A shuffle algebra is a $\mathbb{Z}_{+}$-graded vector space $V=\cup_{i=1}^{\infty}$ such that for any pair $(i,j)$ there exists a collection of operations $*_{\sigma}:V_{i}\otimes V_{j}\rightarrow V_{i+j}$ numbered by $(i,j)$-shuffle permutations $\sigma\in S_{i+j}$ (i.e. $\sigma$ preserves the order of the first $i$ elements and the order of the last $j$ elements) yielding the natural associativity conditions. Enumerative problems for monomial shuffle algebras are in one-to-one correspondence with the pattern avoidance problems for permutations.
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. Both results generalizes the classical results for associative algebras. The first homological result is a generalization of the Golod-Shafarevich theorem and the second one generalizes the theory of Anick chains.
It seems that most of particular applications we discuss are known to specialists but the general method was definitely not known. We hope that it will simplify a lot of work in this area.
It is not hard to see that shuffle algebras form an interesting class of binary shuffle operads and illustrates quite well the importance of the latter notion.
Abstract: 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.
It is now well-known that applications of the operad theory in general (and, in particular, to verifications of the Koszul property) are really difficult in particular computations. There was no known ``arithmetic'' of operations similar to the arithmetic of integers or polynomials (by an ``arithmetic'' we mean the usual notion of divisibility). The good analogue of multiplication for the operadic data is the composition of operations. But the action of the symmetric groups on the entries of operations contradicts with any possible functorial definition of divisibility. This paper contains a solution to this problem using the notion of Shuffle operads. The key idea is to forget about a certain part of the action of the symmetric group. In spite of being a very simple idea, it allowed us to introduce a theory of monomials, their divisibility and compatible orderings of monomials for operads. Summarizing these notions, we came up with the notion of Grobner bases for operads. Grobner bases is a remarkable technical tool initiated in the commutative algebra setting by Buchberger which allows one to solve systems of equations with many unknowns. The theory of Grobner bases for operads made it possible to provide a unified proof of the existing computational results in the field as well as to prove some new results. It is clear that there are many topics that can be successfully approached by these new methods.
This article consists of several parts. We have proved the following:
For any one-generated commutative Koszul algebra $A$ we explain the natural isomorphism between the algebra of Syzygies of $A$ and the Lie algebra cohomology of the $H=H^{\bullet}(L_{\geq 2},\mathbb{C})$, where $L_{\geq 2}$ is the graded Lie subalgebra of the graded Lie super-algebra $L=\oplus_{n=1}^{\infty}L_{n}$ Koszul dual to $A$. We prove that the isomorphism identifies the natural associative algebra structures on $R$ and $H$ coming from their Koszul and Chevalley DGA resolutions respectively. Moreover, it identifies the Higher Massey products in these algebras as well.
For subcanonically embedded $X$ a Frobenius algebra structure on the syzygies is constructed.
All theory was motivated by considering the syzygies for the coordinate algebra of projective variety $X=G/P$ embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie group $G$. We illustrate the results by several examples including the computation of syzygies for the Plucker embeddings of grassmannians $Gr(2,N)$.
It might be that most of particular examples are known for specialists from commutative algebra. However, I like the methods we use for the proofs and I believe that these methods have more powerful application compare to what we have done so far.
Syzygies of some quadratic varieties and their connection with the cohomology of Lie algebras pdf (Russian), TeX (Russian), description, hide
Published in Russian Math. Surveys 61 (2006), no. 5, 990--99
This paper is a short review of the method of computations of syzygies via Lie algebra cohomology presented in details in the joint paper.
Recall that, the Orlik--Solomon algebra $OS(n)$ is a super-commutative associative algebra with odd generators $x_{ij}$, $1\le i\ne j\le n$, and relations \begin{gather*} x_{ij}-x_{ji}=0, \\ x_{ij}^2=0, \quad x_{ij}x_{jk}+x_{jk}x_{ki}+x_{ki}x_{ij}=0. \end{gather*} The Orlik--Solomon algebra $OS(n)$ is isomorphic to the cohomology algebra of the complement of the arrangement $A_{n-1}$ over the complex numbers. In particular it's dimension is equal to $n!$.
The main results of this paper deal with the following generalizations of these algebras suggested by Boris Feigin and Anatol Kirillov. Both our algebras and the ``diagonal coinvariants'' of Haiman coincide with certain quotients of Kirillov's algebras, and this could possibly lead to some new interpretation of diagonal coinvariants. The double Orlik--Solomon algebra $OS_2(n)$ is an associative super-commutative algebra with odd generators $x_{ij}$, $y_{ij}$, $1\le i\ne j\le n$, and relations \begin{gather*} x_{ij}-x_{ji}=y_{ij}-y_{ji}=0,\\ x_{ij}x_{jk}+x_{jk}x_{ki}+x_{ki}x_{ij}=0,\\ x_{ij}y_{jk}+x_{jk}y_{ki}+x_{ki}y_{ij}+y_{ij}x_{jk}+y_{jk}x_{ki}+y_{ki}x_{ij}=0,\\ y_{ij}y_{jk}+y_{jk}y_{ki}+y_{ki}y_{ij}=0,\\ x_{ij}^2=x_{ij}y_{ij}=y_{ij}^2=0. \end{gather*} Main result looks as follows: $dim(OS_2(n))=(n+1)^{n-1}$. The latter number coincides with the dimension of the space of "diagonal coinvariants".
We proved that the union of algebras $OS_2(n)$ form a cooperad. Moreover, we found the isomorphism of the corresponding operad with the Bihamiltonian operad. In particular, this imply not only the knowledge of the dimension of $OS_2(n)$ but also the precise formulas for characters. These characters has been obtained in our previous work, on bihamiltonian operad.
Abstract: We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.
Recall that the series of characters of symmetric groups form a symmetric function. $SL_2$ grading gives an additional parameter to these functions. First, we show how Koszul duality implies the relation for symmetric functions. Namely, the symmetric functions corresponding to the characters of Koszul dual operads should be inverse to each other (up to minor correction of signs) with respect to the plethystic composition. Second, we present a formula for the inverse symmetric function generalizing the Moebius inversion formula.
We proved that the operad of two compatible Lie brackets is Koszul. The koszul dual operad has trivial action of symmetric groups and, therefore, it is easy to write the corresponding symmetric function. Then we use our inversion formula and simplify the result in order to find particular characters for the action of symmetric group.
I insist that the method we suggest is quite general and may be applied in many other situations.
Lie algebra of formal vector fields extended by formal $\mathbf{g}$-valued functions. pdf (Russian), TeX (Russian), description, hide
Published in Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 335 (2006), 205--230;
translation in J. Math. Sci. (N. Y.) 143 (2007), no. 1, 2816--2830,
This paper contains the computation of the cohomology ring of the infinite-dimensional Lie algebra $W_n\ltimes \mathbf{g}\otimes\mathcal{O}_n$ of formal vector fields extended by formal $\mathbf{g}$-valued functions. If $\mathbf{g}$ is a Lie algebra of a compact group $G$ then the corresponding infinite-dimensional Lie algebra $W_n\ltimes \mathbf{g}\otimes\mathcal{O}_n$ is the Lie algebra of infinitesimal deformations of formal trivializations of a $G$-bundle near the point. Based on the ideas from Formal geometry we present a construction of characteristic classes of a $G$-bundle via the Lie algebra cohomology of the desired infinite-dimensional Lie algebra.
Lie algebra of formal vector fields that preserve the foliation structure. pdf (Russian), TeX (Russian), description, hide
Russian: VINITI RAS, N1376-B2006, -- 2006, -- 38pp.; Preprint in Russian ITEP-TH-09/07
The main result of this paper concerns the homology computation of the Lie algebra of formal vector fields on the $n$--dimensional plane (denoted for further by $W_n$). We compute the cohomology with coefficients in symmetric powers of the coadjoint representation.
Theorem For all $k\ge1$ the relative cohomology of the Lie algebra $W_n$ (relative to the subalgebra of linear vector fields) with coefficients in $k$-th symmetric power of coadjoint representation vanishes everywhere except the degree $2n$. The description of absolute cohomology in terms of $\mathsf{gl}_n$-invariants looks as follows: $$H^{i}(W_n;S^{k}W_n^{*}) = \left\{ \begin{array}{l} {[S^{n+k}\mathsf{gl}_n]^{\mathsf{gl}_n}\otimes [\Lambda^{i-2n}(\mathsf{gl}_n)]^{\mathsf{gl}_n}, {\mbox{ if }} 2n\leq i\leq n^2+2n,}\\ {0, {\mbox{ otherwise. }}} \end{array} \right. $$
Recall that the algebra of $\mathsf{gl}_n$--invariants in the symmetric algebra $S^{\bullet}(\mathsf{gl}_{n})$ and in the exterior algebra $\Lambda^{\bullet}(\mathsf{gl}_n)$ are known to coincide with the free symmetric (respectively skew-symmetric) algebra with $n$ generators of degrees $1,2,\ldots, n$ (respectively generators of degrees $1,3,5,\ldots,2n-1$). In particular, we can identify the $2n$'th cohomology $H^{2n}(W_n;S^{k}W_n^{*})$ with the subspace of polynomials of degree $n+k$ in the polynomial algebra $\Bbbk[x_1,x_2,\ldots,x_n]$ (the generator $x_i$ is supposed to have degree $i$).
The conjectural answer in the aforementioned theorem was stated informally in early 70's by B.L.Feigin, D.B.Fuchs and I.M.Gelfand after their quite complicated solution for a particular case of first symmetric power of coadjoint representation ([GFF]). Later on, in 1989 the answer was stated (without any proof) by B.Feigin,B.Tsygan while they use the desired Lie algebra cohomology for the proof of local Riemann-Roch theorem. In 2003 V.Dotsenko ([D]) provides a computation of the dual problem but only for the case $n=1$. Namely, he computed the homology of the Lie algebra of polynomial vector fields on the line with coefficients in symmetric powers of adjoint representation. We suggest the uniform method which does not involve many computations with invariants and covers the case of all positive $n$ and all symmetric powers at the same time.
Moreover, at the same time we compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag of foliations. The latter cohomology are known to be responsible for characteristic classes of flags of foliations. Their construction is given via formal geometry and generalizes the construction of a Godbillon-Vey class.
This preprint was my first attempt to find a useful method to prove Koszulness property for operads. I formulated an interesting definition and have shown a difference between algebras and operads while using the language of distributive lattices.
However, I do not know so far any application of the theory of distributive lattices for operads that may not be attacked by the theory of Grobner Bases (I introduced later). This is why I never published this paper.
This draft contains the computation of the series of characters of Symmetric group for wheeled operads. We give a functional relation for these series for wheeled Koszul dual operads. In particular we proved that none of the possible wheeled completion of the Poisson operad may be wheeled Koszul. This disproves the conjecture of Markl-Merkulov-Shadrin that the wheeled completion of a Koszul operad is wheeled Koszul.
This draft contains a definition of a properad which may be used in order to define the theory of Grobner bases for dioperads. Just plug in this definition into the framework we have suggested for operads.
This draft contains a description by hand of the infinite-dimensional multigraded complex known under the name Bernstein-Gelfand-Gelfand-Felder resolution. The grading is given by affine Weyl group and each graded component consists of the baby Verma module (the finite-dimensional Verma module for a quantum group at the root of unity).
This draft is a conceptual explanation of the existance of BGG duality for Weyl modules and relationship with Macdonald polynomials.
To be finished soon.
This course was given at ETH and was based on the textbook of Weibel. I believe that the list of exercises (accompanied with all definitions) may be useful for someone who wants to learn homological algebra.
The aim of the course was to explain the meaning of the $\infty$-structures and to show different proofs of formality theorem proved by Kontsevich.
My phd
(all materials in Russian) Formal geometry and algebraic invariants of geometric objects. phd - pdf, phd - TeX, review - pdf, review TeX,