On the quadratic dual of the Fomin-Kirillov algebras

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 $n \geq 2$. 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 $n \geq 2$. We also observe that $\mathcal{E}_n^!$ is not prime for $n \geq 3$. By a result of Roos, $\mathcal{E}_n$ is not Koszul for $n \geq 3$, so neither is $\mathcal{E}_n^!$ for $n \geq 3$. 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 $\leq 1$ for each $n \geq 2$, 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.