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## A quasi-coherent description of the the category of D-mod(Gr_GL(n))

Cornell University
,
2018.

Finkelberg M. V., Braverman A.

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr_GL(n)) of (all) D-modules on the affine Grassmannian of the group GL(n) as the category of ind-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a weaker version of this conjecture for the case n=2.

Publication based on the results of:

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2019 Vol. 23 No. 1 P. 75-166

This is a companion paper of [Part II]. We study Coulomb branches
of unframed and framed quiver gauge theories of type ADE. In the
unframed case they are isomorphic to the moduli space of based rational maps from P^1 to the flag variety. In the framed case they are
slices in the affine Grassmannian and their generalization. In ...

Added: September 28, 2019

Bondal Alexey, Rosly Alexei, / IPMU. Series IPMU11-0117 "IPMU11-0117". 2011.

We construct a twist-closed enhancement of the derived category of coherent sheaves on a smooth compact complex-analytic manifold by means of DG-category of dbar-superconnections. ...

Added: October 30, 2013

Bondal A. I., Bodzenta-Skibinska A., Advances in Mathematics 2018 Vol. 323 P. 226-278

Given a relatively projective birational morphism f : X → Y
of smooth algebraic spaces with dimension of fibers bounded
by 1, we construct tilting relative (over Y) generators TX,f
and S_X,f in D^b(X). We develop a piece of general theory of
strict admissible lattice filtrations in triangulated categories
and show that D^b(X) has such a filtration L where the ...

Added: May 2, 2018

Antipov M., Звонарёва А. О., Journal of Mathematical Sciences 2014 Vol. 202 No. 3 P. 333-345

In this paper, all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 are described, using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application, all two-term tilting complexes over a Brauer star algebra are described and their endomorphism rings are ...

Added: December 25, 2018

Elagin A. D., Lunts V., Schnürer O., Moscow Mathematical Journal 2020 Vol. 20 No. 2 P. 277-309

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, thereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated ...

Added: May 11, 2020

Zürich : European Mathematical Society Publishing house, 2012

The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic ...

Added: October 14, 2013

Michael Finkelberg, Feigin E., Reineke M., Kyoto Journal of Mathematics 2017 Vol. 57 No. 2 P. 445-474

We study the connection between the affine degenerate Grassmannians in type A, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type GL(n) and identify it with semi-infinite orbit closure of type A_{2n-1}. We show that principal quiver Grassmannians for the ...

Added: May 10, 2017

Krylov V., Functional Analysis and Its Applications 2018 Vol. 52 No. 2 P. 113-133

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let $\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the integrable crystals $\mathbf{B}^{G}(\lambda),\ \lambda\in\Lambda^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\mathbf{p}_{\lambda_{1},\lambda_{2}}\colon \mathbf{B}^{G}(\lambda_{1}) \otimes \mathbf{B}^{G}(\lambda_{2}) \rightarrow \mathbf{B}^{G}(\lambda_{1}+\lambda_{2})\cup\{0\}$ ...

Added: September 11, 2018

Alexey Bondal, Kavli Institute for the Physics and Mathematics of the Universe News 2011 Vol. 14 P. 4-9

Дается взгляд на развитие идей гомологической алгебры и их приложений к алгебраической геометрии. Описывается связь с зеркальной симметрией и предлагается гомотопическая интерпретация категории производных категорий. ...

Added: October 14, 2013

Braverman A., Michael Finkelberg, Moscow Mathematical Journal 2013 Vol. 13 No. 2 P. 233-265

This is the third paper in a series which describes a conjectural analogue of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The present paper is dedicated to the description of the conjectural analogue of the convolution diagram for the double affine Grassmannian and affine zastava. ...

Added: September 18, 2013

Finkelberg Michael, Fujita R., Representation Theory 2021 Vol. 25 P. 67-89

The convolution ring of loop rotation equivariant K-homology of the affine Grassmannian of GL(n) was identified with
a quantum unipotent cell of the loop group of SL(2) by Cautis and Williams. We identify the basis formed by
the classes of irreducible equivariant perverse coherent sheaves with the dual
canonical basis of the quantum unipotent cell. ...

Added: January 29, 2021

Katzarkov L. V., Karzhemanov I., / Cornell University. Series math "arxiv.org". 2023. No. 2310.13319.

Added: October 31, 2023

Braverman A., Michael Finkelberg, Ginzburg V. et al., Compositio Mathematica 2021 Vol. 157 No. 8 P. 1724-1765

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of GL(N − 1, C[[t]])-equivariant perverse sheaves on the affine Grassmannian of GLN . We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and ...

Added: July 22, 2021

Michael Finkelberg, Krylov V., Mirkovic I., Journal of Topology 2020 Vol. 13 No. 2 P. 683-729

Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups
and consider the corresponding semi-infinite orbits in the affine Grassmannian Gr G . We prove
Simon Schieder’s conjecture identifying his bialgebra formed by the top compactly supported
cohomology of the intersections of opposite semi-infinite orbits with U (n ∨ ) (the universal
enveloping ...

Added: March 19, 2020

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2019 Vol. 23 No. 2 P. 253-344

We consider the
morphism from the variety of triples introduced in our previous paper to the
affine Grassmannian. The direct image of the dualizing complex is a
ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in our previous paper work for an arbitrary commutative
ring object.
The second purpose of this ...

Added: November 12, 2019

Michael Finkelberg, Kamnitzer J., Pham K. et al., Advances in Mathematics 2018 Vol. 327 P. 349-389

We study a coproduct in type A quantum open Toda lattice
in terms of a coproduct in the shifted Yangian of sl2. At
the classical level this corresponds to the multiplication of
scattering matrices of euclidean SU(2) monopoles. We also
study coproducts for shifted Yangians for any simply-laced
Lie algebra. ...

Added: February 21, 2018

Guseva L., / Cornell University. Series math "arxiv.org". 2018.

We construct a full exceptional collection of vector bundles in the bounded derived category of
coherent sheaves on the Grassmannian IGr(3,8) of isotropic 3-dimensional subspaces in a symplectic vector
space of dimension 8. ...

Added: October 19, 2018

Elagin Alexey, Lunts Valery, Schnürer O., / Cornell University. Series arXiv "math". 2018.

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated ...

Added: December 1, 2018

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148

In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...

Added: May 17, 2017