On the quadratic dual of the Fomin-Kirillov algebras

Abstract

We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) ${\mathcal{E}}_n^{!}$ of the Fomin-Kirillov algebras ${\mathcal{E}}_n$; these algebras are connected $\mathbb{N}$-graded and are defined for $n\ge 2$. We establish that the algebra ${\mathcal{E}}_n^{!}$ is module-finite over its center (so, satisfies a polynomial identity), is Noetherian, and has Gelfand-Kirillov dimension $\lfloor n/2\rfloor$ for each $n\ge 2$. We also observe that ${\mathcal{E}}_n^{!}$ is not prime for $n\ge 3$. By a result of Roos, ${\mathcal{E}}_n$ is not Koszul for $n\ge 3$, so neither is ${\mathcal{E}}_n^{!}$ for $n\ge 3$. Nevertheless, we prove that ${\mathcal{E}}_n^{!}$ is Artin-Schelter (AS-)regular if and only if $n=2$, and that ${\mathcal{E}}_n^{!}$ is both AS-Gorenstein and AS-Cohen-Macaulay if and only if $n=2,3$. We also show that the depth of ${\mathcal{E}}_n^{!}$ is $\le 1$ for each $n\ge 2$, conjecture we have equality, and show this claim holds for $n=2,3$. Several other directions for further examination of ${\mathcal{E}}_n^{!}$ are suggested at the end of this article.

v1: 25 pages