On the quadratic dual of the Fomin–Kirillov algebras

Abstract

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