Odd Khovanov's arc algebra

TL;DR

This paper introduces an odd variant of Khovanov's arc algebra, linking it to the odd cohomology of Springer varieties and demonstrating its associative structure through twisting.

Contribution

It constructs the odd arc algebra, extends its center to include anticommuting elements, and proves its associative twisting, advancing the understanding of algebraic structures related to Springer varieties.

Findings

The odd arc algebra is isomorphic to the odd cohomology of $(n,n)$-Springer varieties.

The center extension includes anticommuting elements.

The odd arc algebra can be twisted into an associative algebra.

Abstract

We construct an odd version of Khovanov's arc algebra $H^n$. Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the $(n,n)$-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.