arXiv subjects

Sergey Arkhipov

Publications and source records attributed to Sergey Arkhipov.

At least 19 recordsLinked to original sources

arXiv2025

Equivariant K-theory of the space of partial flags

We use Drinfeld style generators and relations to define an algebra $\mathfrak{U}_n$ which is a ``$q=0$'' version of the affine quantum group of $\mathfrak{gl}_n.$ We then use the convolution product on the equivariant $K$-theory of varieties of pairs of partial flags in a $d$-dimensional vector space $V$ to define affine $0$-Schur algebras ${\mathbb S}_0^{\operatorname{aff}}(n,d)$ and to prove that for every $d$ there exists a surjective homomorphism from $\mathfrak{U}_n$ to ${\mathbb S}_0^{\operatorname{aff}}(n,d).$

math.RT↗

arXiv2023

$A_{\infty}$-structures in monoidal DG categories and strong homotopy unitality

We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong homotopy unitality and bimodule homotopy unitality to work in this level of generality. For a strong homotopy unital $A_{\infty}$-algebra we construct Free-Forgetful homotopy adjunction, its Kleisli category, and its derived category of modules. Analogous constructions for $A_{\infty}$-coalgebras require bicomodule homotopy counitality. We define homotopy adjunction for $A_{\infty}$-algebra and $A_{\infty}$-coalgebra and show such pair to be derived module-comodule equivalent. As an application, we obtain the notions of an $A_{\infty}$-monad and of an enhanced exact monad. We also show that for any adjoint triple $(L,F,R)$ of functors between enhanced triangulated categories the adjunction monad $RF$ and the adjunction comonad $LF$ are derived module-comodule equivalent.

math.CT↗

arXiv2023

Equivariant sheaves on loop spaces

Let $X$ be an affine, smooth, and Noetherian scheme over $\mathbb{C}$ acted on by an affine algebraic group $G$. Applying the technique developed in Arkhipov and Ørsted (2018a, 2018b), we define a dg-model for the derived category of dg-modules over the dg-algebra of differential forms $Ω_X$ on $X$ equivariant with respect to the action of a derived group scheme $(G ,Ω_G )$. We compare the obtained dg-category with the one considered in Arkhipov and Kanstrup (2015) given by coherent sheaves on the derived Hamiltonian reduction of $T^* X$.

math.RT↗

arXiv2021

From polytopes to operads and back

For a directed polytope, we construct a colored operad whose Poincare-Hilbert series encodes certain operations on the cellular complex of the polytope. We conjecture that for a class of short polytopes the constructed operads are Koszul and self-dual. We verify the conjecture for simplices, polygons, and products thereof.

math.KT↗

arXiv2021

A note on a Holstein construction

We clarify details and fill certain gaps in the construction of a canonical Reedy fibrant resolution for a constant simplicial DG-category due to Holstein.

math.AT↗

arXiv2020

Homotopy characters as a homotopy limit

For a Hopf DG-algebra corresponding to a derived algebraic group, we compute the homotopy limit of the associated cosimplicial system of DG-algebras given by the classifying space construction. The homotopy limit is taken in the model category of DG-categories. The objects of the resulting DG-category are Maurer-Cartan elements of $\operatorname{Cobar}(A)$, or 1-dimensional $A_\infty$-comodules over $A$. These can be viewed as characters up to homotopy of the corresponding derived group. Their tensor product is interpreted in terms of Kadeishvili's multibraces. We also study the coderived category of DG-modules over this DG-category.

math.AT↗

arXiv2020

Perverse sheaves on affine flags and Langlands dual group

The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.

math.RT↗

arXiv2019

Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories

We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield-Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.

math.CT↗

arXiv2019

Homotopy limits in the category of dg-categories in terms of $\mathrm{A}_{\infty}$-comodules

In this paper, we apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (2016) and Arkhipov and Poliakova (2018), as well as our own results on homotopy ends (Arkhipov and Ørsted 2018), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of $\mathrm{A}_{\infty}$-comodules, proving a conjecture by Block, Holstein, and Wei (2017) in the process.

math.CT↗

arXiv2018

Colored DG-operads and homotopy adjunction for DG-categories

Generalizing the approach to pseudo monoidal DG-categories as certain colored non-symmetric DG-operads, we introduce a certain relaxed notion of a category enriched in DG-categories. We construct model structures on the category of colored non-symmetric DG-operads and on the category of DGCat-enriched categories with a fixed set of objects. This allows us to talk about strong homotopy maps in both settings. We discuss the notion of a strong homotopy monad in a DG-category and a notion of strong homotopy adjunction data for two DG-functors.

math.CT↗

arXiv2015

Demazure descent and representations of reductive groups

We introduce the notion of Demazure descent data on a triangulated category C and define the descent category for such data. We illustrate the definition by our basic example. Let G be a reductive algebraic group with a Borel subgroup B. Demazure functors form Demazure descent data on the derived category of Rep(B) and the descent category is equivalent to the derived category of Rep(G).

math.RT↗

arXiv2015

Quasi-coherent Hecke category and Demazure Descent

Let G be a reductive algebraic group with a Borel subgroup B. We define the quasi-coherent Hecke category for the pair (G,B). For any regular Noetherian G-scheme X we construct a monoidal action of the Hecke category on the derived category of B-equivariant quasi-coherent sheaves on X. Using the action we define the Demazure Descent Data on the latter category and prove that the Descent category is equivalent to the derived category of G-equivariant sheaves on X.

math.RT↗

arXiv2015

Equivariant Matrix Factorizations and Hamiltonian reduction

Let $X$ be a smooth scheme with an action of an algebraic group $G$. We establish an equivalence of two categories related to the corresponding moment map $μ: T^*X \to Lie(G)^*$ - the derived category of G-equivariant coherent sheaves on the derived fiber $μ^{-1}(0)$ and the derived category of $G$-equivariant matrix factorizations on $T^*X \times Lie(G)$ with potential given by $μ$.

math.RT↗

arXiv2015

Braid group actions on matrix factorizations

Let $X$ be a smooth scheme with an action of a reductive algebraic group $G$ over an algebraically closed field $k$ of characteristic zero. We construct an action of the extended affine Braid group on the $G$-equivariant absolute derived category of matrix factorizations on the Grothendieck variety times $T^*X$ with potential given by the Grothendieck-Springer resolution times the moment map composed with the natural pairing.

math.RT↗

arXiv2007

2-gerbes and 2-Tate spaces

We construct a central extension of the group of automorphisms of a 2-Tate vector space viewed as a discrete 2-group. This is done using an action of this 2-group on a 2-gerbe of gerbel theories. This central extension is used to define central extensions of double loop groups.

math.CT↗

arXiv2005

Toric arc schemes and quantum cohomology of toric varieties

We describe the quantum cohomology ring of a toric Fano variety $X$ in terms of the usual topological cohomology ring for an auxiliary infinite-dimensional scheme. This scheme is a part of an algebro-geometric model for the universal cover of the space of free loops with values in $X$.

math.AG↗

arXiv2004

Quantum Groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra which have played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p-adic reductive group, while the other is in terms of equivariant K-theory of a complex (Steinberg) variety for the dual group. Our equivalence (2)-(3) may be viewed as a `categorification' of the isomorphism between the corresponding two geometric realizations of the fundamental polynomial representation of the affine Hecke algebra. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on representation theory of Kac-Moody algebras). It also gives way to proving Humphreys' conjectures on tilting U_q(g)-modules, as will be explained in a separate paper.

math.RT↗

arXiv2001

Algebraic construction of contragradient quasi-Verma modules in positive characteristic

In the present paper we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive chararacteristic called quasi-Verma modules. We provide a purely algebraic construction of the global Grothendieck-Cousin complex corresponding to the standard line bumdle ${\mathcal L}(λ)$ on the Flag variety of the algebraic group stratified by Schubert cells. We prove that the complex consists of direct sums of quasi-Verma modules for the highest weights of the form $w\cdotλ$ for various elements $w$ of the Weyl group.

math.AG↗