Quantum group coproducts and universality under scalar extensions

TL;DR

This paper characterizes when bialgebra, Hopf algebra, and coalgebra products are finite-dimensional or preserved under scalar extensions over various fields, clarifying conditions for universality and correcting previous errors.

Contribution

It provides new characterizations of finite-dimensionality and preservation of algebraic structures under scalar extensions, correcting earlier misconceptions.

Findings

Finite-dimensionality of products depends on field size and factor dimensions.

Scalar extension functors preserve algebraic structures under specific field extension conditions.

Characterization of algebraic and finite field extensions based on preservation of coalgebra and bialgebra products.

Abstract

We characterize the families of bialgebras or Hopf algebras over fields for which the product in the corresponding category is finite-dimensional, answering a question of M. Lorenz: if the ground field is infinite then bialgebra or Hopf products are finite-dimensional precisely when the factors are, with at most one of dimension $>1$; over finite fields the necessary and sufficient condition is instead that factors be finite-dimensional with at most finitely many of dimension $>1$; finally, these statements hold for coalgebras as well, provided the family is finite. We also characterize (a) finite field extensions as precisely those whose underlying scalar extension functor preserves coalgebra, or bialgebra, or Hopf algebra products (correcting an error in the literature); (b) algebraic field extensions as those along which finite coalgebra (bialgebra, Hopf algebra) products are preserved; and (c) again algebraic field extensions as precisely those which intertwine cofree coalgebras on vector spaces, or cofree bialgebras (Hopf algebras) on algebras.