On the quadratic dual of the Fomin–Kirillov algebras
On the quadratic dual of the Fomin–Kirillov algebras
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DOI：
10.1090/tran/7781
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发表时间：
2018-06
影响因子：
1.3
通讯作者：
Chelsea M. Walton;James J. Zhang
中科院分区：
文献类型：
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作者：
Chelsea M. Walton;James J. Zhang
我们研究环理论和同调性质的二次对偶（或Koszul对偶）En！\mathcal {E} n^！Fomin-Kirillov代数E n \mathcal {E}_n ;这些代数是连通的N \mathbb {N} -分次的，并且定义为n ≥ 2 n \geq 2。我们建立了代数E n！\mathcal {E} n^！是在其中心上有限的模（因此满足多项式恒等式），是诺特的，并且对于每个n ≥ 2 n \geq 2，有Gelfand-Kirillov维数n/ 2 n\lfloor n/2 \rfloor。我们还观察到，E n！\mathcal {E} n^！对于n ≥ 3 n \geq 3不是素数。通过鲁什的一个结果，En\mathcal {E}_n对n ≥ 3 n \geq 3不是Koszul，所以En也不是Koszul！\mathcal {E} n^！对于n ≥ 3 n \geq 3 .然而，我们证明，E n！\mathcal {E} n^！是Artin-Schelter（AS-）正则的当且仅当n = 2 n=2，且E n！\mathcal {E} n^！是AS-Gorenstein和AS-Cohen-Macaulay当且仅当n = 2，3。我们还表明，E n！\mathcal {E} n^！是≤ 1 \leq 1对于每个n ≥ 2 n \geq 2，猜想我们有平等，并表明这一主张适用于<mml：math xmlns：mml=“http：//
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 <mml:math xmlns:mml="http://