Heisenberg algebra and a graphical calculus

TL;DR

This paper introduces a graphical calculus for Heisenberg algebras using planar diagrams, leading to a new categorical framework that captures algebraic structures and provides explicit bases for morphism spaces.

Contribution

It develops a novel graphical calculus involving biadjoint functors and affine Hecke algebras, creating a categorical model for the Heisenberg algebra with explicit bases.

Findings

Constructed a monoidal category with Grothendieck ring containing an integral form of the Heisenberg algebra

Developed bases for morphism spaces between products of generating objects

Connected diagrammatic calculus to algebraic structures in representation theory

Abstract

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 Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of vector spaces of morphisms between products of generating objects in this category.