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On the quadratic dual of the Fomin-Kirillov algebras. (English) Zbl 1471.16051

Summary: 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 J.-E. Roos [Prog. Math. 172, 385–389 (1999; Zbl 0962.16022)], \(\mathcal{E}_n\) 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.

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16P40 Noetherian rings and modules (associative rings and algebras)
16P90 Growth rate, Gelfand-Kirillov dimension
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)

Citations:

Zbl 0962.16022

Software:

GBNP
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References:

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