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Open Access Publications from the University of California

Scholarly Works (5 results)

  • Article
  • Peer Reviewed
UC Riverside Previously Published Works (2016)
In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for nonhereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.
    • Thesis
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    UC Riverside Electronic Theses and Dissertations (2020)

    We study connections between quantum cluster algebras and the double canonical bases of subalgebras of the Heisenberg and Drinfeld double associated to a quantized Borel subalgebra of $\mathfrak{sl}_3$. We show that the Heisenberg double has a finite type quantum cluster algebra structure for which the set of quantum cluster monomials is equal to the double canonical basis. Furthermore, we identify an affine quantum cluster algebra structure on parabolic subalgebras of the Drinfeld double and prove that all quantum cluster variables belong to the double canonical basis. Finally, we identify an infinite subset of quantum clusters for which the quantum cluster monomials are contained in the double canonical basis.

      Cover page: Cluster Structures in Double Canonical Bases
      • Article
      • Peer Reviewed
      UC Riverside Previously Published Works (2015)
      We generalize the decomposition of Uq(g) introduced by A. Joseph in [5] and link it, for g semisimple, to the celebrated computation of central elements due to V. Drinfeld [2]. In that case, we construct a natural basis in the center of Uq(g) whose elements behave as Schur polynomials and thus explicitly identify the center with the ring of symmetric functions.
        • Thesis
        • Peer Reviewed
        UC Riverside Electronic Theses and Dissertations (2015)

        We study the structure of a Kashiwara crystal of simply-laced Cartan type \(\cd\) under an automorphism \(\sigma\), via a process known as folding.

        We define the category of \(\sigma\)-foldable crystals, which is a monoidal category and admits a monoidal functor into the category of \(\cd^{\sigma\vee}\)-crystals.

        Various properties of a foldable crystal---normality, Weyl group action, \emph{etc.}---can be transferred to its quotient modulo the \(\sigma\)-action.

        On the other hand, the quotient of a foldable highest-weight crystal contains a new type of crystal, which we call multi-highest-weight.

        We consider multi-highest-weight crystals \(B\) obtained as foldings of Kashiwara Littelmann crystals \(B(\lambda)\).

        The structure of the highest-weight set of \(B\) is explained by certain subsets of the Weyl group we call the balanced parabolic quotients; in many cases the latter parameterizes a generating set for the highest-weight elements.

        A balanced parabolic quotient relates the branching rules, Demazure crystals, and \(\sigma\)-action on \(B(\lambda)\), and is enumerated by a forest graph with self-similar components.

          Cover page: Combinatorics of Crystal Folding
          • Article
          • Peer Reviewed
          UC Riverside Previously Published Works (2015)
          The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations if one starts with a principal quantum seed. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel, and M. Kontsevich.
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