An Exceptional Splitting of Khovanov's Arc Algebras in Characteristic 2

TL;DR

This paper demonstrates a unique algebraic splitting of Khovanov's arc algebra in characteristic 2, revealing a new structural decomposition not present over the integers.

Contribution

It introduces an associative algebra $ ilde{H}_n$ such that $H_n$ splits as $ ilde{H}_n[x]/(x^2)$ in characteristic 2, providing new insights into algebraic structures in this setting.

Findings

Khovanov's arc algebra $H_n$ is isomorphic to $ ilde{H}_n[x]/(x^2)$ over characteristic 2.

No such algebraic splitting exists over the integers.

The result extends to bimodules associated with planar tangles.

Abstract

We show that there is an associative algebra $\widetilde{H}_n$ such that, over a base ring $R$ of characteristic 2, Khovanov's arc algebra $H_n$ is isomorphic to the algebra $\widetilde{H}_n[x]/(x^2)$. We also show a similar result for bimodules associated to planar tangles and prove that there is no such isomorphism over $\mathbb{Z}$.