String diagrams and categorification

@article{Savage2018StringDA,
  title={String diagrams and categorification},
  author={Alistair Savage},
  journal={arXiv: Representation Theory},
  year={2018}
}
  • Alistair Savage
  • Published 18 June 2018
  • Mathematics
  • arXiv: Representation Theory
These are lectures notes for a mini-course given at the conference Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras, and Categorification in June 2018. The goal is to introduce the reader to string diagram techniques for monoidal categories, with an emphasis on their role in categorification. 
Presentations of diagram categories.
We describe the planar rook category, the rook category, the rook-Brauer category, and the Motzkin category in terms of generators and relations. We show that the morphism spaces of these categories
Presentations of linear monoidal categories and their endomorphism algebras.
We give the definition of presentations of linear monoidal categories. Our main result is that given a presentation of a linear monoidal category, we can produce a presentation of the same category
Embedding Deligne's category $\mathrm{Rep}(S_t)$ in the Heisenberg category.
We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category $\mathrm{Rep}(S_t)$, to the Heisenberg category. We show that the induced map on
Affine oriented Frobenius Brauer categories
To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer categeory. We define natural actions of these categories on categories of
Rewriting modulo isotopies in pivotal linear (2,2)-categories
The algebra and machine representation of statistical models
TLDR
This dissertation takes steps toward digitizing and systematizing two major artifacts of data science, statistical models and data analyses, by designing and implementing a software system for creating machine representations of data analyses in the form of Python or R programs.

References

SHOWING 1-10 OF 21 REFERENCES
A categorification of quantum sl(n)
To an arbitrary root datum we associate a2-category. For root datum corresponding to sl.n/ we show that this 2-category categorifies the idempotented form of the quantum enveloping algebra.
Current algebras and categorified quantum groups
TLDR
It is shown that 2-representations defined using categories of modules over cyclotomic quotients of KLR-algebras correspond to local (or global) Weyl modules.
Quantum affine wreath algebras
To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type
Heisenberg categorification and Hilbert schemes
Given a finite subgroup G of SL(2,C) we define an additive 2-category H^G whose Grothendieck group is isomorphic to an integral form of the Heisenberg algebra. We construct an action of H^G on
W-ALGEBRAS FROM HEISENBERG CATEGORIES
The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra $W_{1+\infty }$ . This induces an action of $W_{1+\infty }$ on the center of
Heisenberg algebra and a graphical calculus
A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose
A basis theorem for the affine oriented Brauer category and its cyclotomic quotients
The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a
Hecke algebras, finite general linear groups, and Heisenberg categorification
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category,
On the definition of Heisenberg category
We revisit the definition of the Heisenberg category of central charge k. For central charge -1, this category was introduced originally by Khovanov, but with some additional cyclicity relations
...
1
2
3
...