Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0812.4864

Groupoidification Made Easy

John C. Baez·Alexander E. Hoffnung·Christopher D. Walker
View original source ↗Read source document ↗Export metadata ↓

Abstract

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements.

Explore related subjects

math.QAmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

John C. Baez, Alexander E. Hoffnung, Christopher D. Walker. 2008-12-30. Groupoidification Made Easy. https://arxiv.org/abs/0812.4864

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Birational Equivalences for Kac--Moody Borel Enveloping Algebras

A Coxeter ordering of the simple roots of a finite-rank Kac--Moody algebra determines a finite family of commuting real-root vectors. We prove that $U^{\geq0}(\mathfrak g)$ is birationally equivalent to $Z\otimes\mathbb A_n$, where $Z$ is the residual Coxeter centralizer, by identifying the Coxeter localization $U^{\geq0}(\mathfrak g)[\mathbf X^{-1}]$ with $Z\otimes\mathbb A_n[\mathbf x^{-1}]$. For symmetrizable Cartan matrices the residual algebra is generated by finite Coxeter windows and is finitely presented. For the generic quantum Borel with torus dual to the root lattice, we prove the analogous birational equivalence.

math.QA↗
arXiv2026

A diagrammatic presentation for every pivotal pointed fusion category

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,ω,π)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category.

math.QA↗
arXiv2026

The Kazhdan-Lusztig category of $\mathfrak{osp}_{1|2n}$ at irrational levels

We prove the Kazhdan-Lusztig correspondence for the Lie superalgebra $\mathfrak{osp}_{1|2n}$ at irrational levels, that is, we show the category $\mathrm{KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ of finite-length even ordinary modules for the affine vertex operator superalgebra of $\mathfrak{osp}_{1|2n}$ at level $k \in \mathbb{C} \setminus \mathbb{Q}$ is braided tensor equivalent to the category of finite-dimensional even weight modules for the quantum group of $\mathfrak{osp}_{1|2n}$ at parameter $q = e^{πi/(2k+2n+1)}$. We also prove that ${\rm KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ is braided tensor equivalent to the category ${\rm KL}_\ell^{\rm ns}(\mathfrak{so}_{2n+1})$ of finite-length ordinary modules with non-spinorial top level for the affine vertex operator algebra of $\mathfrak{so}_{2n+1}$ at level $\ell$ such that $ \frac{1}{\ell+ 2n-1} = \frac{1}{2k+2n+1} + 1 \ \ ({\rm mod}\ 2\mathbb Z).$ Consequently, by gluing vertex operator (super)algebras via tensor categories, we construct a few new families of simple conformal vertex (super)algebras, including the mixed kernel VOAs that were the missing ingredient for proving certain Feigin-Frenkel type dualities in previous work of the first-named author with Linshaw, Nakatsuka, and Sato.

math.QA↗