A categorical reconstruction of crystals and quantum groups at $q=0$

TL;DR

This paper explores the structure of crystals and quantum groups at q=0, showing that crystal bases cannot be realized as coalgebras over pointed sets, but can be modeled by a new bialgebra conjecturally related to Lusztig's quantum group.

Contribution

It classifies crystal bases as coalgebras over a comonadic functor and constructs a bialgebra over integers linking crystals to quantum groups at infinity.

Findings

No coalgebra in pointed sets corresponds to crystal bases.

Constructed a bialgebra over Z with crystals as comodules.

Conjectured connection to Lusztig's quantum group at v=∞.

Abstract

The quantum co-ordinate algebra $A_{q}(\mathfrak{g})$ associated to a Kac-Moody Lie algebra $\mathfrak{g}$ forms a Hopf algebra whose comodules are precisely the $U_{q}(\mathfrak{g})$ modules in the BGG category $\mathcal{O}_{\mathfrak{g}}$. In this paper we investigate whether an analogous result is true when $q=0$. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over $\mathbb{Z}$ whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig's quantum group at $v = \infty$.