On the dimension of the Fomin-Kirillov algebra and related algebras

TL;DR

This paper proves that the Fomin-Kirillov algebra $ ext{E}_m$ is infinite dimensional for all $m \\geq 6$, resolving a longstanding conjecture and using advanced algebraic techniques.

Contribution

It establishes the infinite dimensionality of $ ext{E}_m$ for all $m \\geq 6$, a key open problem in the study of these algebras.

Findings

$ ext{E}_m$ is infinite dimensional for all $m \\geq 6$

Uses braided differential calculus and Hopf algebra integrals

Resolves a well-known conjecture in algebra

Abstract

In 1999, Fomin-Kirillov introduced the quadratic algebras $\mathcal{E}_m$ in terms of generators and relations which are the universal quadratic cover of the algebra generated by divided difference operators $\partial_{ij}$ acting on the polynomial ring $\mathbf{k}[x_1,\ldots,x_m]$. These algebras are mostly important due to their relations to Schubert calculus and geometry and to the general framework of quantum groups and Nichols algebras. Fomin and Kirillov asked about the dimension of $\mathcal{E}_m$. In this paper, we prove that $\mathcal{E}_m$ is infinite dimensional for all $m\geq 6$ which was a well-known conjecture. The techniques we use rely on braided differential calculus as developed by Liu and Bazlov as well as on the notion of integrals for Hopf algebras as introduced by Sweedler.