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## Two-term partial tilting complexes over Brauer tree algebras

Journal of Mathematical Sciences. 2014. Vol. 202. No. 3. P. 333-345.

Antipov M., Звонарёва А. О.

Translator: Antipov M.

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 computed.

Positselski L., Arkhipov S., Rumynin D., Basel : Birkhauser/Springer, 2010

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories ...

Added: March 19, 2013

Feigin E., Makedonskyi I., Orr D., Advances in Mathematics 2018 Vol. 330 P. 997-1033

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules. ...

Added: September 13, 2018

Cruz Morales J. A., Galkin S., Upper Bounds for Mutations of Potentials / Cornell University. Series math "arxiv.org". 2013. No. 1301.4541.

In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. ...

Added: May 27, 2013

Positselski L., Categorical Bockstein sequences / Cornell University. Series math "arxiv.org". 2014. No. 1404.5011.

We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The procedure allows, e.g., to recover the abelian/exact category of modular representations of a finite group from the exact category of its l-adic ...

Added: April 22, 2014

Feigin E., Makedonskyi I., Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration / Cornell University. Series math "arxiv.org". 2014. No. 1407.6316.

The Cherednik-Orr conjecture expresses the t\to\infty limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases. ...

Added: August 10, 2014

Alexander I. Efimov, Derived categories of Grassmannians over integers and modular representation theory / Cornell University. Series math "arxiv.org". 2014.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category D^b(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GL_k. This in particular implies existence of a full exceptional collection, which is ...

Added: February 2, 2015

Bondal Alexey, Rosly Alexei, Derived categories for complex-analytic manifolds / . 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

Braverman A., Michael Finkelberg, Twisted zastava and q-Whittaker functions / Cornell University. Series math "arxiv.org". 2014.

In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces. ...

Added: February 5, 2015

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., Dobrovolska G., Michael Finkelberg, Gaiotto-Witten superpotential and Whittaker D-modules on monopoles / Cornell University. Series math "arxiv.org". 2014.

Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of ...

Added: February 3, 2015

Makhlin I., Selecta Mathematica, New Series 2015

We exploit the idea that the character of an irreducible finite dimensional $\mathfrak{gl}_n$-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits ...

Added: September 29, 2014

Levin A., Olshanetsky M., Zotov A., Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations / Cornell University. Series math "arxiv.org". 2014.

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the ...

Added: January 23, 2015

Feigin B. L., Miwa T., Jimbo M. et al., Branching rules for quantum toroidal $\mathfrak{gl}_n$ / Cornell University Library. 2013. No. 1309.2147.

We construct an analog of the subalgebra $U\mathfrak{gl}(n)\otimes U\mathfrak{gl}(m)\subset U\mathfrak{gl}(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra. ...

Added: April 24, 2014

Michael Finkelberg, Schechtman V., Microlocal approach to Lusztig's symmetries / Cornell University. Series math "arxiv.org". 2014.

We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between the Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves. ...

Added: January 30, 2015

Bershtein M., Gavrylenko P., Marshakov A., Twist-field representations of W-algebras, exact conformal blocks and character identities / . 2017. No. 1705.00957.

We study twist-field representations of the W-algebras and generalize the construction of the corresponding vertex operators to D- and B-series. We demonstrate how the computation of characters of such representations leads to the nontrivial identities involving lattice theta-functions. We propose a construction of their exact conformal blocks, which for D-series express them in terms of ...

Added: May 4, 2017

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

Braverman A., Michael Finkelberg, Nakajima H., Instanton moduli spaces and W-algebras / Cornell University. Series math "arxiv.org". 2014.

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras"). ...

Added: January 30, 2015

Bezrukavnikov R., Finkelberg M. V., Wreath Macdonald polynomials and categorical McKay correspondence (with Appendices by Ivan Losev, Vadim Vologodsky) / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Rovinsky M., On semilinear representations of the infinite symmetric group / Cornell University. Series math "arxiv.org". 2014.

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of ...

Added: September 17, 2014

Finkelberg M. V., Braverman A., A quasi-coherent description of the the category of D-mod(Gr_GL(n)) / Cornell University. Series arXiv "math". 2018.

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 ...

Added: December 3, 2018

Elagin Alexey, Lunts Valery, Schnürer O., Smoothness of Derived Categories of Algebras / 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

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

[б.и.], 2016

Added: September 26, 2016

Pirkovskii A. Y., Известия РАН. Серия математическая 2012 Т. 76 № 4 С. 65-124

We prove the equation w.dg A = w.db A for every nuclear Fréchet–Arens–Michael algebra A of finite weak bidimension, where w.dg A is the weak global dimension and w.db A is the weak bidimension of A. Assuming that A has a projective bimodule resolution of finite type, we establish the estimate dg A ≤ db ...

Added: September 19, 2012