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 $n \geq 2$. We establish that the algebra $\mathcal {E}_n^!$ is module finite over its center (and thus 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 that we have equality, and show that 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.