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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the quadratic dual of the Fomin–Kirillov algebras
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by Chelsea Walton and James J. Zhang PDF
Trans. Amer. Math. Soc. 372 (2019), 3921-3945 Request permission

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.
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Additional Information
  • Chelsea Walton
  • Affiliation: Department of Mathematics, The University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
  • MR Author ID: 879649
  • Email: notlaw@illinois.edu
  • James J. Zhang
  • Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
  • MR Author ID: 314509
  • Email: zhang@math.washington.edu
  • Received by editor(s): July 6, 2018
  • Published electronically: February 11, 2019
  • Additional Notes: The first author was partially supported by a research fellowship from the Alfred P. Sloan foundation, and by the U.S. National Science Foundation grants #DMS-1663775, 1903192. This work was completed during her visits to the University of Washington–Seattle.
    The second author was partially supported by U.S. National Science Foundation grant #DMS-1700825.
    Part of this work was completed during the authors’ attendance at the “Quantum Homogeneous Spaces” workshop at the International Centre for Mathematical Sciences in Edinburgh, Scotland; the authors appreciate the institution staff for their hospitality and assistance during these stays.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 3921-3945
  • MSC (2010): Primary 16W50, 16P40, 16P90, 16E65
  • DOI: https://doi.org/10.1090/tran/7781
  • MathSciNet review: 4009423