On the quadratic dual of the fomin–Kirillov algebras

Chelsea Walton, James J. Zhang

Research output: Contribution to journalArticlepeer-review


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

Original languageEnglish (US)
Pages (from-to)3921-3945
Number of pages25
JournalTransactions of the American Mathematical Society
Issue number6
StatePublished - Sep 15 2019


  • Depth
  • Fomin–Kirillov algebra
  • Gelfand–Kirillov dimension
  • Homological conditions
  • Noetherian
  • Quadratic (Koszul) dual

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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