Digital quantum groups

Abstract

We find and classify all bialgebras and Hopf algebras or “quantum groups” of dimension ≤4 over the field F2={0,1}. We summarize our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras and, among them, 25 Hopf algebras, with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover self-dual and resembles a digital version of uq(sl2). We also find a unique self-dual Hopf algebra in one anyonic variable x4 = 0. For all our Hopf algebras, we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or “universal R-matrix” structures on our Hopf algebras. These induce solutions of the Yang–Baxter or braid relations in any representation.