Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

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).

Download Full-text

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501915 ◽

2013 ◽

Vol 12 (04) ◽

pp. 1250191

Author(s):

XIAOLAN YU ◽

YINHUO ZHANG

Keyword(s):

Spectral Sequence ◽

Hopf Algebras ◽

Type A ◽

Serre Spectral Sequence ◽

Nichols Algebra ◽

Cartan Type ◽

Dynkin Diagrams ◽

Full Structure ◽

Pointed Hopf Algebras ◽

Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

Download Full-text

A LERAY SPECTRAL SEQUENCE FOR NONCOMMUTATIVE DIFFERENTIAL FIBRATIONS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887813500151 ◽

2013 ◽

Vol 10 (05) ◽

pp. 1350015 ◽

Cited By ~ 9

Author(s):

EDWIN BEGGS ◽

IBTISAM MASMALI

Keyword(s):

Spectral Sequence ◽

Sheaf Cohomology ◽

Left Module ◽

Graded Algebras ◽

Serre Spectral Sequence ◽

Differential Graded Algebras ◽

Zero Curvature ◽

Total Differential ◽

Leray Spectral Sequence ◽

Special Case

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.

Download Full-text