TY - JOUR

T1 - On the quadratic dual of the fomin–Kirillov algebras

AU - Walton, Chelsea

AU - Zhang, James J.

N1 - Funding Information:
Received by the editors July 6, 2018. 2010 Mathematics Subject Classification. Primary 16W50, 16P40, 16P90, 16E65. Key words and phrases. Fomin–Kirillov algebra, Gelfand–Kirillov dimension, homological conditions, quadratic (Koszul) dual, Noetherian, depth. 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.
Publisher Copyright:
© 2019 American Mathematical Society.

PY - 2019/9/15

Y1 - 2019/9/15

N2 - 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.

AB - 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.

KW - Depth

KW - Fomin–Kirillov algebra

KW - Gelfand–Kirillov dimension

KW - Homological conditions

KW - Noetherian

KW - Quadratic (Koszul) dual

UR - http://www.scopus.com/inward/record.url?scp=85075157448&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075157448&partnerID=8YFLogxK

U2 - 10.1090/tran7781

DO - 10.1090/tran7781

M3 - Article

AN - SCOPUS:85075157448

SN - 0002-9947

VL - 372

SP - 3921

EP - 3945

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -