About

Ken Goodearl is an Affiliate Professor in the Department of Mathematics at the University of Washington. He holds a PhD from the University of Washington, obtained in 1971. His fields of interest include Noncommutative Algebra. Further details about his research focus or key contributions are not provided on the page.

Research topics

• Mathematics
• Pure mathematics
• Physics
• Quantum mechanics
• Combinatorics

Selected publications

• Cluster algebra structures on Poisson nilpotent algebras

  Memoirs of the American Mathematical Society · 2023 · 15 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics

  Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

  DOI

• Integral quantum cluster structures

  Duke Mathematical Journal · 2021 · 14 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Quantum mechanics

  We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q±1/2]. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra g and Weyl group element w, the dual canonical form Aq(n+(w))Z[q±1] of the corresponding quantum unipotent cell has the property that Aq(n+(w))Z[q±1]⊗Z[q±1]Z[q±1/2] is isomorphic to a quantum cluster algebra over Z[q±1/2] and to the corresponding upper quantum cluster algebra over Z[q±1/2].

  DOI

• Integral quantum cluster structures

  arXiv (Cornell University) · 2020 · 5 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Physics

  We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${\mathbb{Z}}[q^{\pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${\mathfrak{g}}$ and Weyl group element $w$, the dual canonical form $A_q({\mathfrak{n}}_+(w))_{\mathbb{Z}[q^{\pm 1}]}$ of the corresponding quantum unipotent cell has the property that $A_q( {\mathfrak{n}}_+(w))_{\mathbb{Z}[q^{\pm 1}]} \otimes_{\mathbb{Z}[q^{ \pm 1}]} {\mathbb{Z}}[ q^{\pm 1/2}]$ is isomorphic to a quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$ and to the corresponding upper quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$.

  DOI

Recent grants

• Research in Ring Theory and Noncommutative Algebraic Geometry

  NSF · $170k · 2004–2009

• Research in Ring Theory and Noncommutative Algebraic Geometry

  NSF · $151k · 2008–2013

• Research in Ring Theory and Noncommutative Algebraic Geometry

  NSF · $224k · 2016–2023

Frequent coauthors

• Ken A. Brown

  51 shared

• T. H. Lenagan

  Maxwell Institute for Mathematical Sciences

  36 shared

• Pere Ara

  32 shared

• Enrique Pardo

  21 shared

• Edward S. Letzter

  Temple University

  15 shared

• Milen Yakimov

  Northeastern University

  14 shared

• Stéphane Launois

  University of Kent

  12 shared

• Friedrich Wehrung

  Normandie Université

  10 shared

Labs

• Math AI LabPI

Similar researchers at University of Washington

• Sarah Jaewon Leeh-151
• Jan Hansenh-130
• Linda Simonsenh-120
• Anna Waldh-119
• Jesse Bloomh-117