nLab quantum group

Idea

The notion of quantum group refers to various objects which are deformations of (algebras of functions on) groups, but still have very similar properties to (algebras of functions on) groups, and in particular to semisimple Lie groups. Most important are the Hopf algebras deforming the function algebras on semisimple Lie groups or to the enveloping algebras of Kac-Moody Lie algebras.

Overview

It is a common experience in representation theory that a number of mathematical structures behaves very similarly to algebraic or Lie groups. After the impetus of the theory of quantum integrable systems, mainly the work of Leningrad’s school of mathematical physics around 1980, several mathematicians (including Drinfeld, Manin, Woronowicz, Jimbo, Faddeev–Reshetikhin–Takhtajan) found, in different formalisms, major series of examples which are mostly noncommutative noncocommutative Hopf algebras and which deform enveloping algebras of (semisimple) Lie algebras, or algebras of functions on the corresponding algebraic groups. These deformations $G_q$ depend on a parameter $q$ (sometimes one prefers a formal parameter $h$ with $q = e^{h}$), which may be taken as belonging to the ground field, but also being formal (transcendental over the ground field). A peculiar case is when the parameter $q$ of the deformation is an $l$-th root of unity; the remaining cases are usually called generic $q$.

The representation theory for these ‘quantum’ examples is highly developed; in fact many phenomena in the representation theory of semisimple Lie algebras (e.g. canonical bases) were discovered first as a limiting case of constructions in the quantum case, which become degenerate in the classical case (the principle that quantization removes degeneracy). While representations for generic $q$ parallel classical ones, the theory at roots of unity is peculiar and related to the representation theory of affine Lie algebras; the quantum groups at roots of unity as algebras have big centers.

Nowadays, both the class of examples and the class of formalisms has been extended a lot, hence the term ‘quantum group’ is not a fixed notion but rather a collective term for a rather author-dependent class of group-like objects, most often subclasses or extensions of the concept of Hopf algebras which are sometimes required to belong to families of deformations of their classical counterparts. One of the common features is that if we forget the group-like features, the examples belong to the class of noncommutative spaces (see noncommutative geometry).

Mathematically better defined are notions (sometimes equated by various authors with the class of quantum groups) like quasitriangular Hopf algebras, quantum matrix groups? (quantum linear groups, more general FRT-algebras and Majid’s $A(R)$ where $R$ is a quantum Yang-Baxter equation), quantized enveloping algebras, quantum function algebras, compact matrix pseudogroups, Kac algebras, Yangians etc. The representations of quasitriangular Hopf algebras form braided monoidal categories, which are in main examples related to the mathematics of Iwahori–Hecke algebras, braid groups, knot theory, finite group Chern–Simons theory and Wess–Zumino–Novikov–Witten theory of CFT. One should note that in the classical limit quantum function algebras give not simply (functions on) algebraic (or Lie) groups but also a compatible (= multiplicative) Poisson structure giving rise to Poisson–Lie or Poisson algebraic groups.

There is an extensive geometric theory of homogeneous spaces for quantum groups and fiber bundles whose structure groups are quantum groups.

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

Related concepts

• quantum algebra

• noncommutative geometry, noncommutative algebraic geometry, deformation quantization

• quantum inverse scattering method, Knizhnik-Zamolodchikov equation, Tannaka duality, Yang-Baxter equation, classical Yang-Baxter equation, quantum Yang-Baxter equation, dynamical Yang-Baxter equation, reflection equation

• Hopf algebra, bialgebra, gebra, Yangian, RTT algebra, matrix bialgebra, quantum linear group, quantized function algebra, quantized enveloping algebra

• braided monoidal category, quantum double, Heisenberg double, Yetter-Drinfeld module

• quantum homogeneous space, quantum flag manifold, quantum symmetric pair, Hopf-Galois extension, noncommutative principal bundle

• Poisson Lie group, dressing action

• quantum 2-group

• quantum topology

References

• Vladimir Drinfeld, Quantum groups, in: A. Gleason (ed.) Proceedings of the 1986 International Congress of Mathematics 1 (1987) 798–820 [pdf]

  expanded version:

  Journal of Soviet Mathematics 41 (1988) 898–915 [doi:10.1007/BF01247086]

• Tjark Tjin, An introduction to quantized Lie groups and algebras, Int. J. Mod. Phys. A 7 (1992) 6175–6213 [arXiv:hep-th/9111043, doi:10.1142/S0217751X92002805]

• Christian Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer 1995 (doi:10.1007/978-1-4612-0783-2, webpage, errata pdf)

• Shahn Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

• Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.

• Brian Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

• N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i analiz 1, 178 (1989) (Russian), English translation in Leningrad Math. J. 1.

• Arun Ram, A survey of quantum groups: background, motivation, and results, in: Geometric analysis and Lie theory in mathematics and physics, A. Carey and M. Murray eds., Australian Math. Soc. Lecture Notes Series 11, Cambridge Univ. Press 1997, pp. 20-104. pdf

• Pavel Etingof, O. Schiffmann, Lectures on Quantum Groups, Lectures in Math. Phys., International Press (1998).

• Pavel Etingof, Igor Frenkel, Lectures on representation theory and Knizhnik-Zamolodchikov equations

• A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.

• A. Joseph, Quantum groups and their primitive ideals, Springer 1995.

• Ross Street, Quantum groups : a path to current algebra, Cambridge Univ. Press 2007

• L. I. Korogodski, Ya. S. Soibelman, Algebras of functions on quantum groups I, Math. Surveys and Monographs 56, AMS 1998.

• A. Varchenko, Hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics, Vol. 21, World Scientific (1995)

• George Lusztig, Introduction to quantum groups

• V. Chari, A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994

• Bangming Deng, Jie Du, Brian Parshall, Jianpan Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs 150, Amer. Math. Soc. 2008. xxvi+759 pp. MR2009i:17023)

• Tom Bridgeland, Quantum groups via Hall algebras of complexes, Annals of Mathematics 177:2 (2013) 739–759 (21 pages)

• Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin, Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups, 2020 (pdf)

In relation to hypergeometric functions and the Knizhnik-Zamolodchikov equation:

• Alexander Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv. Ser. in Math. Phys. 21, World Sci. Publ. 1995. x+371 pp. (doi:10.1142/2467)

• V. Tarasov, Alexander Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pp. (arXiv:q-alg/9703044, numdam:AST_1997__246__R1_0)

In the generality of noncartesian internal categories:

• Marcelo Aguiar, Internal categories and quantum groups, PhD thesis, Cornell 1997 (pdf, pdf)

On elliptic quantum groups:

• Hitoshi Konno, Elliptic Quantum Groups, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2405.11193]

See also:

• MathOverflow: q.gr. as relative Drinfeld double, why Drinfeld-Jimbo q.gr., Lusztig perverse sheaves on quiver varieties, canonical bases for extended q.env.algebras, groups-qgroups-and-… (on elliptic case), all posts with quantum group tag

category: noncommutative geometry