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arXiv:2507.22541 · Journal of Physics A Mathematical and Theoretical

Two-dimensional bialgebras and quantum groups: algebraic structures and tensor network realizations

José Garre-Rubio, András Molnár, Germán Sierra

Published Feb 3, 2026• 0 citations
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Abstract

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 [ s u ( 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 projected entangled pair states, 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.

DOI
10.1088/1751-8121/ae414d

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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 [ s u ( 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 projected entangled pair states, 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.
Matrix Product Operators (MPOs) are tensor networks representing operators acting on 1D systems. They model a wide variety of situations, including communication channels with memory effects, quantum cellular automata, mixed states in 1D quantum systems, or holographic boundary models associated to 2D quantum systems. A scenario where MPOs have proven particularly useful is to represent algebras of non-trivial symmetries. Concretely, the boundary of both symmetry protected and topologically ordered phases in 2D quantum systems exhibit symmetries in the form of MPOs. In this paper, we develop a theory of MPOs as representations of algebraic structures. We establish a dictionary between algebra and MPO properties which allows to transfer results between both setups, covering the cases of pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion of pulling-through algebras, which abstracts the minimal requirements needed to define topologically ordered 2D tensor networks from MPO algebras. We show, as one of our main results, that any semisimple pivotal weak Hopf algebra is a pulling-through algebra. We demonstrate the power of this framework by showing that they can be used to construct Kitaev's quantum double models for Hopf algebras solely from an MPO representation of the Hopf algebra, in the exact same way as MPO symmetries obtained from fusion categories can be used to construct Levin-Wen string-net models, and to explain all their topological features; it thus allows to describe both Kitaev and string-net models on the same formal footing.
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on M(H)=Hcop▶ ⁣ ⁣ ⁣◃HM(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HM(H)=Hcop▶◃H as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to D(H)D(H)D(H). Given a finite-dimensional Hopf algebra HHH, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop▶ ⁣ ⁣ ⁣◃HH^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HHcop▶◃H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
Abstract Kitaev’s lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid’s bicrossproduct quantum group, related by semidualisation or ‘quantum Born reciprocity’ to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z 2 topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin- S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how local perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices.
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double. We propose a new version based on Majid's bicrossproduct quantum group. Given a Hopf algebra HHH, we show that a triangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop▶ ⁣ ⁣ ⁣◃HH^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HHcop▶◃H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. We obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
This dissertation investigates well-known (quantum) combinatorial structures, such as (quantum) designs and (quantum) orthogonal Latin squares through the lenses of matrix algebras, group theory, and category theory. A central focus is placed on perfect tensors that are examined using representation theory and quasi-orthogonal subalgebras of matrix algebras. In particular, it is shown that the existence of a 2-unitary in dimension d-square is equivalent to the existence of four mutually quasi-orthogonal subalgebras of a d-square dimensional matrix algebra that are each isomorphic to a d-dimensional matrix algebra. This correspondence is further interpreted in terms of groups and their representations, relating the existence of perfect tensors to the existence of a group with a d-square dimensional irreducible representation and four subgroups on which the respective character “factorises“. A search algorithm implemented in the algebra software GAP is presented that was used to find examples of such groups. Furthermore, additional construction schemes for other dimensions, leading to 2-unitary complex Hadamard matrices, are developed using doubly perfect bi-unimodular sequences. Using the same ansatz, an analytical resolution of Euler’s 36 officers problem is provided, which differs from the earlier constructions of perfect tensors in dimension 36 via computer-assisted methods. Beyond this, the thesis introduces an abstract notion of categorical designs formulated via arrow categories. When applied to the categories Mat(N) and CP[FHilb], this framework produces a category-theoretic description of balanced incomplete block designs and of quantum designs, respectively. The construction generalises earlier definitions of quantum designs and extends it to a broader concept that can also be interpreted as combinatorial superoperators. These new concepts are used to define a category of mutually unbiased bases and a category of so-called combinatorial quantum channels.