scholarly journals Braiding via geometric Lie algebra actions

2012 ◽  
Vol 148 (2) ◽  
pp. 464-506 ◽  
Author(s):  
Sabin Cautis ◽  
Joel Kamnitzer
Keyword(s):  
Lie Algebra ◽  
Braid Group ◽  
Group Actions ◽  
Derived Categories ◽  
Flag Varieties ◽  
Coherent Sheaves ◽  
Cotangent Bundles ◽  
Partial Flag Varieties

AbstractWe introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

scholarly journals Singular Localization and Intertwining Functors for Reductive Lie Algebras in Prime Characteristic

2006 ◽  
Vol 183 ◽  
pp. 1-55 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Ivan Mirković ◽  
Dmitriy Rumynin
Keyword(s):  
Lie Algebra ◽  
Differential Operators ◽  
Derived Categories ◽  
Central Character ◽  
Basic Step ◽  
Finite Characteristic ◽  
Coherent Sheaves ◽  
Springer Fiber ◽  
Category Of Modules ◽  
Central Characters

In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

scholarly journals Categorification via blocks of modular representations for

2020 ◽  
pp. 1-29
Author(s):  
Vinoth Nandakumar ◽  
Gufang Zhao
Keyword(s):  
Lie Algebras ◽  
Positive Characteristic ◽  
Tensor Products ◽  
Derived Categories ◽  
Modular Representations ◽  
Combinatorial Approach ◽  
Standard Representation ◽  
Coherent Sheaves ◽  
Cotangent Bundles ◽  
Representations Of Lie Algebras

Abstract Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

Stability conditions and braid group actions on affine An quivers

2021 ◽  
pp. 2250174
Author(s):  
Chien-Hsun Wang
Keyword(s):  
Moduli Space ◽  
Braid Group ◽  
Group Actions ◽  
Stability Conditions ◽  
Quadratic Differentials ◽  
Spherical Subgroup ◽  
Type Formula ◽  
Affine Type

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

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