Quantum group coproducts and universality under scalar extensions
Alexandru Chirvasitu
TL;DR
This paper addresses when products in Hopf algebras, bialgebras, and coalgebras over a field are finite-dimensional, identifying sharp conditions that depend on the field's cardinality and the dimensions of the factors. It develops a detailed analysis of how scalar extension along field extensions interacts with universal constructions, proving that finite extensions precisely correspond to preservation of products (and related limits) in these categories, while algebraic extensions preserve cofree coalgebra and cofree (bialgebra/Hopf) constructions on algebras. The results correct a prior misconception about product preservation under scalar extension and establish a clear dichotomy: finite vs algebraic vs transcendental extensions govern the permanence properties of products, cofree coalgebras, and related structures. The work uses adjunctions, cofree constructions, and Tannakian-style perspectives to rigorously characterize when scalar extension preserves or fails to preserve these universal objects, with consequences for understanding 'quantum group' coproducts under scalar changes.
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.