Two-Dimensional Bialgebras and Quantum Groups: Algebraic Structures and Tensor Network Realizations

TL;DR

This paper develops a framework for defining coalgebra and bialgebra structures on 2D lattices, extending quantum group theory beyond 1D, and explores tensor network realizations of these algebraic structures.

Contribution

It introduces a novel 2D algebraic framework for quantum groups and demonstrates how tensor networks induce such structures in higher dimensions.

Findings

Constructed 2D bialgebras with horizontal and vertical coproducts.

Generalized quantum group $U_q[su(2)]$ to 2D and analyzed $q$-deformed states.

Linked tensor network states to 2D coalgebra structures.

Abstract

We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group $U_q[su(2)]$, for which we construct 2D generalizations of its generators, analyze $q$-deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly PEPS, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.