2018/06/25 by Chelsea Walton, Walton, Chelsea, James J. Zhang +1
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.QA #math.RA
paper · pdf · doi:10.48550/arxiv.1806.09263
v1: 25 pages
arxiv created 2018/06/25 · arxiv updated 2018/06/26
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 (so, satisfies a polynomial identity), is Noetherian, and has Gelfand-Kirillov dimension \lfloor n/2 \rfloor 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 we have equality, and show this claim holds for n =2,3. Several other directions for further examination of En^! are suggested at the end of this article.