# Quantum group
A quantum group is a noncommutative, and usually noncocommutative, [Hopf algebra](https://www.edgepedia.org/hopf-algebra) regarded as a deformation of the algebraic structure of a group or of a universal enveloping algebra of a [Lie algebra](https://www.edgepedia.org/lie-algebra). The term was introduced by [Vladimir Drinfeld](https://www.edgepedia.org/vladimir-drinfeld) at the 1986 International Congress of Mathematicians, together with a supply of new examples, and independently by Michio Jimbo in work on the quantum Yang–Baxter equation[1](https://www.cambridge.org/core/books/quantum-groups/introduction/E536C985B10734CB76EFC18A284D2F62) • [2](https://www.mate.unlp.edu.ar/~ggarcia/encuentros/notas-curso-qg-ha.pdf). There is no single rigorous, universally accepted definition; the term is generally agreed to cover certain one- or multi-parameter deformations of classical objects attached to algebraic groups, such as enveloping algebras of semisimple Lie algebras and algebras of regular functions on those groups[2](https://www.mate.unlp.edu.ar/~ggarcia/encuentros/notas-curso-qg-ha.pdf).
| Key fact | Detail |
|---|---|
| Core object | A Hopf algebra: an algebra equipped with comultiplication \( \Delta \), counit \( \epsilon \), and antipode \( S \)[3](https://encyclopediaofmath.org/wiki/Quantum_groups) |
| Category convention | The category of quantum groups is taken to be dual (opposite) to the category of Hopf algebras[2](https://www.mate.unlp.edu.ar/~ggarcia/encuentros/notas-curso-qg-ha.pdf) |
| Origin of the term | Coined by Drinfeld at the 1986 International Congress of Mathematicians, with new examples; parallel work by Jimbo[1](https://www.cambridge.org/core/books/quantum-groups/introduction/E536C985B10734CB76EFC18A284D2F62) |
| Deformation parameter | A parameter \( q \) plays a role analogous to Planck's constant; at \( q \to 1 \) quantum groups become the classical groups and algebras[4](https://nsr-jinr.ru/article/download/42/34/337) |
| Central example | The Drinfeld–Jimbo algebra \( U_q(\mathfrak{g}) \), a \( q \)-deformation of the universal enveloping algebra of a semisimple Lie algebra \( \mathfrak{g} \)[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf) |
| R-matrix property | In a quasitriangular Hopf algebra the universal R-matrix satisfies the quantum Yang–Baxter equation \( R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12} \)[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf) |
| Root-of-unity behavior | At roots of unity, representation theory loses semisimplicity; suitable quotient categories are modular and yield knot invariants and topological quantum field theories[6](https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs/20) |
## From groups to Hopf algebras
The route from ordinary groups to quantum groups runs through the algebra of functions. A group \( G \) acting on a manifold \( M \) also acts on the algebra of functions \( F(M) \); replacing \( F(M) \) by a general noncommutative algebra leads to the notion of a Hopf algebra as something capable of acting on such a noncommutative space[7](https://maths-magic.ac.uk/courses/2024-2025/magic109).
Concretely, the functor that sends a space to its algebra of functions is an anti-equivalence between spaces and commutative algebras. Translating the group axioms into algebra, the multiplication \( G \times G \to G \) becomes a comultiplication \( \Delta : A \to A \otimes A \), the unit element gives a counit \( \epsilon : A \to k \), and inversion \( g \mapsto g^{-1} \) gives an antipode \( S \). The group axioms are then equivalent to the commutativity of standard diagrams, so that \( (A, \Delta, \epsilon, S) \) is a commutative Hopf algebra[3](https://encyclopediaofmath.org/wiki/Quantum_groups). Because groups and commutative Hopf algebras form categories that are anti-equivalent, it makes sense to regard a quantum group as an object in the category opposite to that of Hopf algebras that need not be commutative.[3](https://encyclopediaofmath.org/wiki/Quantum_groups) For this reason some authors define quantum groups simply as noncommutative and noncocommutative Hopf algebras[2](https://www.mate.unlp.edu.ar/~ggarcia/encuentros/notas-curso-qg-ha.pdf).
Basic examples of Hopf algebras predate the term: the algebra of regular functions on an affine algebraic group is a commutative Hopf algebra, and the group algebra \( \mathbb{C}G \), with \( \Delta(g) = g \otimes g \), \( S(g) = g^{-1} \), and \( \epsilon(g) = 1 \), is a cocommutative one[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf). Universal enveloping algebras of Lie algebras are Hopf algebras as well[8](https://math.ucr.edu/~jdolan/ram1.pdf).
## The Drinfeld–Jimbo quantum groups
The most important nontrivial examples are quantizations of ordinary classical Lie groups and algebras: one deforms either the algebra of functions on a [Lie group](https://www.edgepedia.org/lie-group) or the universal enveloping algebra of a Lie algebra. The deformation is controlled by a parameter \( q \), which plays a role analogous to Planck's constant in quantum mechanics, and in the limit \( q \to 1 \) the quantum group goes over into the classical object[4](https://nsr-jinr.ru/article/download/42/34/337). The [quantized enveloping algebra](https://www.edgepedia.org/quantized-enveloping-algebra) \( U_q(\mathfrak{g}) \) of a complex semisimple Lie algebra \( \mathfrak{g} \) is the central case[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf).
These algebras arose in the study of the [quantum inverse scattering method](https://www.edgepedia.org/quantum-inverse-scattering-method) in integrable systems, in work of the mid-1980s on the quantum [Yang–Baxter equation](https://www.edgepedia.org/yang-baxter-equation) and \( q \)-analogs of enveloping algebras[9](https://arxiv.org/abs/hep-th/0703013). Their representation theory parallels the classical [highest-weight theory](https://www.edgepedia.org/highest-weight-theory) when \( q \) is not a root of unity; for \( U_q(\mathfrak{sl}_2) \) the parallel with the representation theory of \( \mathfrak{sl}_2 \) is explicit, with the universal R-matrix and the Yang–Baxter equation playing a central role[10](https://www.aqualab.unipr.it/appel/phd.html).
**Quasitriangularity and the quantum double.** A Hopf algebra is quasitriangular when it carries a universal R-matrix satisfying the quantum Yang–Baxter equation \( R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12} \)[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf). Drinfeld's quantum double construction \( D(H) \) produces such a structure from any Hopf algebra \( H \), and its representation category is braided[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf). The quantum double, the R-matrix formalism, and the associated braid group quotients such as the Hecke and Birman–Murakami–Wenzl algebras form the R-matrix approach to the theory of quantum groups[4](https://nsr-jinr.ru/article/download/42/34/337).
## Roots of unity and the small quantum group
When \( q \) is a root of unity, the representation theory changes character. Semisimplicity fails: Weyl modules are not always simple and filtrations by them do not always split[11](https://citeseerx.ist.psu.edu/document?doi=941f024f2dc32b2d0fc59a5c9250fdf175a6d78c&repid=rep1&type=pdf). In this setting [George Lusztig](https://www.edgepedia.org/george-lusztig) introduced the small quantum group \( u_q(\mathfrak{sl}_2) \), a quotient Hopf algebra of dimension \( \ell^3 \) at a root of unity of order \( \ell \), and generalized the construction to any simple Lie algebra[5](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf).
The root-of-unity theory is what connects quantum groups to topology. For each compact, simple, simply-connected Lie group and each integer level, an appropriate quotient of a subcategory of representations of the corresponding quantum group is a modular tensor category, which produces invariants of links and three-manifolds[6](https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs/20). Stephen F. Sawin, a mathematician at [Fairfield University](https://www.edgepedia.org/fairfield-university), developed this representation theory for all quantum groups at all roots of unity, including Harish–Chandra's theorem, and showed that at each integer level the quotient category is modular, leading to a topological quantum field theory[6](https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs/20).
## Applications
**Knot invariants.** The representation theory of \( U_q(\mathfrak{sl}_2) \) encodes information that, expressed through a knot diagram in a well-defined graphical calculus, produces the [Jones polynomial](https://www.edgepedia.org/jones-polynomial), one of the best-known knot invariants[12](https://scholarship.richmond.edu/honors-theses/975/). The R-matrix formalism gives a general formulation of link and knot invariants[4](https://nsr-jinr.ru/article/download/42/34/337), and the construction of the Jones polynomial from \( U_q(\mathfrak{sl}_2) \) is an example of a topological quantum field theory[7](https://maths-magic.ac.uk/courses/2024-2025/magic109).
**Integrable systems.** Quantum groups and the Yang–Baxter equation generate classes of quantum integrable systems in a systematic way, a connection present from the subject's origins[13](https://www.sciencedirect.com/science/article/pii/0960077994E0102U). Quantum affine algebras, the quantum groups associated to affine Lie algebras, have a representation theory tied to quantum integrable models: transfer-matrix eigenvalues are expressed through Baxter polynomials, and exchange relations in associated cluster algebras correspond to Bethe Ansatz relations[14](https://arxiv.org/abs/2510.06437).
**Physics and beyond.** Quantum groups serve as symmetries of noncommutative spaces, which motivated their study in mathematical physics, including deformations of Lorentz and Poincaré symmetries and Yangian symmetries of scattering amplitudes in supersymmetric [Yang–Mills theory](https://www.edgepedia.org/yang-mills-theory)[4](https://nsr-jinr.ru/article/download/42/34/337) • [9](https://arxiv.org/abs/hep-th/0703013). In condensed matter, the deformed representation theory of quantum groups supplies F-matrices and R-matrices for anyon theories, giving data that in many cases is equivalent to that of [Chern–Simons theory](https://www.edgepedia.org/chern-simons-theory) based on the corresponding Lie group[15](https://academic.oup.com/book/55346/chapter/429680508). Hopf algebras and their representations also appear in topology, number theory, quantum information theory, and, more recently, finance through stochastic calculus[7](https://maths-magic.ac.uk/courses/2024-2025/magic109).
## Classification problems
One of the main open problems in the theory of Hopf algebras is the classification of Hopf algebras of a fixed dimension over an algebraically closed field of characteristic zero; quantum groups enter this program as the distinguished class of examples against which general classifications are tested[2](https://www.mate.unlp.edu.ar/~ggarcia/encuentros/notas-curso-qg-ha.pdf). On the analytic side, a parallel classification effort concerns compact quantum groups of Lie type, meaning quantum groups whose fusion rules and representation dimensions match those of a compact connected Lie group[16](https://arxiv.org/abs/1603.05519).
## References
1. [introduction - quantum groups](https://www.cambridge.org/core/books/quantum-groups/introduction/E536C985B10734CB76EFC18A284D2F62)
2. [Gastón Andrés García (2009). Quantum Groups and Hopf Algebras, XVIII Latin American Algebra Colloquium lecture notes](https://www.mate.unlp.edu.ar/~ggarcia/encuentros/notas-curso-qg-ha.pdf)
3. [Quantum groups, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Quantum_groups)
4. [A. P. Isaev (2025). Quantum groups and Yang–Baxter equations, Natural Science Review 2, 100204, JINR](https://nsr-jinr.ru/article/download/42/34/337)
5. [Pavel Etingof and Mykola Semenyakin. A Brief Introduction to Quantum Groups, Harvard CMSA](https://cmsa.fas.harvard.edu/media/Etinghof_skoltechlect2-1.pdf)
6. [Stephen F. Sawin (2006). Quantum groups at roots of unity and modularity, Journal of Knot Theory and Its Ramifications 15(10), 1245–1277](https://digitalcommons.fairfield.edu/mathandcomputerscience-facultypubs/20)
7. [MAGIC109: Introduction to Hopf algebras and quantum groups, course description](https://maths-magic.ac.uk/courses/2024-2025/magic109)
8. [Arun Ram. Quantum groups: A survey of definitions, motivations, and results, Princeton University](https://math.ucr.edu/~jdolan/ram1.pdf)
9. [Paolo Aschieri (2007). Lectures on Hopf Algebras, Quantum Groups and Twists, arXiv:hep-th/0703013](https://arxiv.org/abs/hep-th/0703013)
10. [Andrea Appel, PhD course Introduction to Quantum Groups and Quantum Loop Algebras, University of Parma](https://www.aqualab.unipr.it/appel/phd.html)
11. [Representations of Quantum Groups at Roots of Unity, CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=941f024f2dc32b2d0fc59a5c9250fdf175a6d78c&repid=rep1&type=pdf)
12. [Greg A. Hamilton. Quantum Groups and Knot Invariants, University of Richmond honors thesis](https://scholarship.richmond.edu/honors-theses/975/)
13. [Origin of quantum group and its application in integrable systems, Chaos, Solitons & Fractals 5(12), 2329–2344 (1995)](https://www.sciencedirect.com/science/article/pii/0960077994E0102U)
14. [David Hernandez (2025). Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models, arXiv:2510.06437](https://arxiv.org/abs/2510.06437)
15. [Steven H. Simon (2023). Quantum Groups (in Brief), in Topological Quantum, Oxford University Press](https://academic.oup.com/book/55346/chapter/429680508)
16. [Towards a classification of compact quantum groups of Lie type, arXiv:1603.05519](https://arxiv.org/abs/1603.05519)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Hopf and quantum algebras › Quantum groups*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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