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The phrase "quantum group" is more or less a synonym for "Hopf algebra" . More precisely, the category of quantum groups is defined in [[#References|[a1]]] to be dual to the category of Hopf algebras. This is natural for the following reason. There is the following general principle: The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763101.png" /> is an anti-equivalence between the category of "spaces" and the category of commutative associative unital algebras, perhaps with some additional structures or properties (this principle becomes a theorem if "space" is understood to be "affine scheme" or "compact topological space" , and "algebra" is understood to mean "C*-algebra" ). So one can translate the definition of a group into the language of algebras: instead of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763102.png" /> with an associative operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763103.png" /> one obtains a commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763104.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763105.png" /> with a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763106.png" />, called comultiplication; the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763107.png" /> gives rise to a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763108.png" />, called co-unit, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631010.png" />, gives rise to a bijective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631011.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631012.png" />, called antipode. The group axioms are equivalent to the commutativity of the following diagrams:
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The phrase "quantum group" is more or less a synonym for "Hopf algebra" . More precisely, the category of quantum groups is defined in [[#References|[a1]]] to be dual to the category of Hopf algebras. This is natural for the following reason. There is the following general principle: The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763101.png" /> is an anti-equivalence between the category of "spaces" and the category of commutative associative unital algebras, perhaps with some additional structures or properties (this principle becomes a theorem if "space" is understood to be "affine scheme" or "compact topological space" , and "algebra" is understood to mean "C*-algebra" ). So one can translate the definition of a group into the language of algebras: instead of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763102.png" /> with an associative operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763103.png" /> one obtains a commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763104.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763105.png" /> with a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763106.png" />, called comultiplication; the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763107.png" /> gives rise to a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763108.png" />, called co-unit, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q0763109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631010.png" />, gives rise to a bijective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631011.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631012.png" />, called antipode. The group axioms are equivalent to the commutativity of the following diagrams:
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631013.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631013.png" /></td> </tr></table>
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A simple class of non-commutative Hopf algebras is formed by the group algebras of non-commutative groups. These Hopf algebras are commutative, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631020.png" /> is contained in the symmetric part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631021.png" />. Essentially, all cocommutative Hopf algebras are group algebras.
 
A simple class of non-commutative Hopf algebras is formed by the group algebras of non-commutative groups. These Hopf algebras are commutative, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631020.png" /> is contained in the symmetric part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631021.png" />. Essentially, all cocommutative Hopf algebras are group algebras.
  
Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631024.png" /> is a commutative ring. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631025.png" /> the associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631026.png" />-algebra with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631028.png" />, and defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631029.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631033.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631036.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631040.png" /> is the number of inversions in the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631042.png" /> has a Hopf algebra structure defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631045.png" /> is the algebra of polynomial functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631046.png" />. So, in the general case it is natural to consider elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631047.png" /> as "functions on the quantized SLn" .
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Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631024.png" /> is a commutative ring. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631025.png" /> the associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631026.png" />-algebra with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631028.png" />, and defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631029.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631033.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631036.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631040.png" /> is the number of inversions in the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631042.png" /> has a Hopf algebra structure defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631045.png" /> is the algebra of polynomial functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631046.png" />. So, in the general case it is natural to consider elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631047.png" /> as "functions on the quantized SLn" .
  
 
The quantized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631048.png" /> is one of the simplest quantum groups which appear naturally in the theory of quantum integrable systems and, especially, in the quantum inverse-scattering method [[#References|[a2]]]. The development of this method has led to the following quantization technique for constructing non-commutative non-cocommutative Hopf algebras. It is natural to construct them as deformations of commutative Hopf algebras. If a non-commutative deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631049.png" /> of a commutative Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631050.png" /> is given, then a Poisson bracket on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631051.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631053.png" /> is the deformation parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631054.png" /> means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631055.png" />) and is compatible with comultiplication. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631056.png" /> is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631057.png" /> and then try to quantize it, i.e. to construct a Hopf algebra deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631058.png" /> which induces the given Poisson bracket on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631059.png" />.
 
The quantized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631048.png" /> is one of the simplest quantum groups which appear naturally in the theory of quantum integrable systems and, especially, in the quantum inverse-scattering method [[#References|[a2]]]. The development of this method has led to the following quantization technique for constructing non-commutative non-cocommutative Hopf algebras. It is natural to construct them as deformations of commutative Hopf algebras. If a non-commutative deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631049.png" /> of a commutative Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631050.png" /> is given, then a Poisson bracket on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631051.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631053.png" /> is the deformation parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631054.png" /> means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631055.png" />) and is compatible with comultiplication. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631056.png" /> is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631057.png" /> and then try to quantize it, i.e. to construct a Hopf algebra deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631058.png" /> which induces the given Poisson bracket on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631059.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310100.png" /> is a finite-dimensional simple Lie algebra (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), then the algebra of regular functions on the corresponding simply-connected algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310101.png" /> is isomorphic to the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310102.png" /> generated by the matrix elements of the finite-dimensional representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310103.png" />. Therefore the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310104.png" /> generated by the matrix elements of the finite-dimensional representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310105.png" /> can be considered as the algebra of functions on a certain quantization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310106.png" />. For instance, the quantized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310107.png" /> (cf. above) can be obtained in this way.
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310100.png" /> is a finite-dimensional simple Lie algebra (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), then the algebra of regular functions on the corresponding simply-connected algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310101.png" /> is isomorphic to the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310102.png" /> generated by the matrix elements of the finite-dimensional representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310103.png" />. Therefore the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310104.png" /> generated by the matrix elements of the finite-dimensional representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310105.png" /> can be considered as the algebra of functions on a certain quantization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310106.png" />. For instance, the quantized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310107.png" /> (cf. above) can be obtained in this way.
  
There is an important notion of a quasitriangular Hopf algebra. This is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310108.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310109.png" /> is a Hopf algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310110.png" /> is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310111.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310114.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310115.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310116.png" /> is the opposite comultiplication and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310119.png" /> are defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310120.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310121.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310124.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310125.png" /> is a quasitriangular Hopf algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310126.png" /> satisfies the quantum Yang–Baxter equation (cf. also [[Yang–Baxter equation|Yang–Baxter equation]]), i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310127.png" />. It is known (cf. [[#References|[a1]]], §13) that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310128.png" /> is a finite-dimensional simple Lie algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310129.png" /> has a canonical quasitriangular structure, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310130.png" /> is an infinite-dimensional Kac–Moody algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310131.png" /> has an "almost quasitriangular" structure.
+
There is an important notion of a quasitriangular Hopf algebra. This is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310108.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310109.png" /> is a Hopf algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310110.png" /> is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310111.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310114.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310115.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310116.png" /> is the opposite comultiplication and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310119.png" /> are defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310120.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310121.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310124.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310125.png" /> is a quasitriangular Hopf algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310126.png" /> satisfies the quantum Yang–Baxter equation (cf. also [[Yang–Baxter equation|Yang–Baxter equation]]), i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310127.png" />. It is known (cf. [[#References|[a1]]], §13) that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310128.png" /> is a finite-dimensional simple Lie algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310129.png" /> has a canonical quasitriangular structure, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310130.png" /> is an infinite-dimensional Kac–Moody algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310131.png" /> has an "almost quasitriangular" structure.
  
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310132.png" /> is a quasitriangular Hopf algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310134.png" /> is a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310135.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310136.png" /> satisfies the quantum Yang–Baxter equation. There is an inverse construction (cf. [[#References|[a6]]], [[#References|[a7]]]), which goes back to the quantum inverse-scattering method: to a matrix solution of the quantum Yang–Baxter equation satisfying a non-degeneracy condition there corresponds a Hopf algebra. Without this condition one can only construct an associative bi-algebra (the difference between a Hopf algebra and an associative bi-algebra is that in the second case there may be no antipode). This bi-algebra is generated by elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310138.png" />, with defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310139.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310142.png" /> is the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310143.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310144.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310145.png" />.
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310132.png" /> is a quasitriangular Hopf algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310134.png" /> is a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310135.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310136.png" /> satisfies the quantum Yang–Baxter equation. There is an inverse construction (cf. [[#References|[a6]]], [[#References|[a7]]]), which goes back to the quantum inverse-scattering method: to a matrix solution of the quantum Yang–Baxter equation satisfying a non-degeneracy condition there corresponds a Hopf algebra. Without this condition one can only construct an associative bi-algebra (the difference between a Hopf algebra and an associative bi-algebra is that in the second case there may be no antipode). This bi-algebra is generated by elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310138.png" />, with defining relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310139.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310142.png" /> is the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310143.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310144.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310145.png" />.
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.G. Drinfel'd,   "Quantum groups" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''1''' , Amer. Math. Soc. (1987) pp. 798–820</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.D. Faddeev,   "Integrable models in (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310152.png" />)-dimensional quantum field theory" , ''Lectures in Les Houches, 1982'' , '''Session 39''' , Elsevier (1984) pp. 563–608</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Jimbo,   "Quantum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310153.png" />-matrix for the generalized Toda system" ''Comm. Math. Phys.'' , '''102''' (1986) pp. 537–547</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Jimbo,   "A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310154.png" />-difference analogue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310155.png" /> and the Yang–Baxter equation" ''Letters Math. Phys.'' , '''10''' (1985) pp. 63–69</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> V.G. Drinfel'd,   "Hopf algebras and the quantum Yang–Baxter equation" ''Soviet Math. Dokl.'' , '''32''' (1985) pp. 254–258 ''Dokl. Akad. Nauk SSSR'' , '''283''' : 5 (1985) pp. 1060–1064</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> V.V. Lyubashenko,   "Hopf algebras and vector symmetries" ''Russian Math. Surveys'' , '''41''' : 5 (1986) pp. 153–154 ''Uspekhi Mat. Nauk'' , '''41''' : 5 (1986) pp. 185–186</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L.D. Faddeev,   N.Yu. Reshetikhin,   L.A. Takhtayan,   "Quantization of Lie groups and Lie algebras" ''Algebra and Analysis'' , '''1''' : 1 (1989) pp. 178–206 (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> N.Yu. Reshetikhin,   "Quasitriangular Hopf algebras and invariants of tangles" ''Algebra and Analysis'' , '''1''' : 2 (1989) pp. 169–188 (In Russian)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> V.F.R. Jones,   "A polynomial invariant for knots via von Neumann algebras" ''Bulletin Amer. Math. Soc.'' , '''12''' (1985) pp. 103–112</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S.L. Woronowich,   "Compact matrix pseudogroups" ''Comm. Math. Phys.'' , '''111''' (1987) pp. 613–665</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S.L. Woronowich,   "Twisted SU(2) group. An example of a noncommutative differential calculus" ''Publ. RIMS'' , '''23''' (1987) pp. 117–181</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L.L. Vaksman,   Ya.S. Soibelman,   "Function algebra on the quantum group SU(2)" ''Funct. Anal. Appl.'' , '''22''' : 3 (1988) pp. 170–181 ''Funksional. Anal. Prilozhen.'' , '''22''' : 3 (1988) pp. 1–14</TD></TR><TR><TD valign="top">[a13a]</TD> <TD valign="top"> G.I. Kac,   "Ring groups and the duality principle I" ''Trans. Moscow Math. Soc.'' , '''12''' (1963) pp. 291–339 ''Trudy Moskov. Mat. Obshch.'' , '''12''' (1963) pp. 295–301</TD></TR><TR><TD valign="top">[a13b]</TD> <TD valign="top"> G.I. Kac,   "Ring groups and the duality principle II" ''Trans. Moscow Math. Soc.'' , '''13''' (1965) pp. 94–126 ''Trudy Moskov. Mat. Obshch.'' , '''13''' (1965) pp. 84–113</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> G.I. Kac,   L.I. Vainerman,   "Nonunimodular ring groups and Hopf–von Neumann algebras" ''Math. USSR Sb.'' , '''23''' (1974) pp. 185–214 ''Mat. Sb.'' , '''94''' : 2 (1974) pp. 194–225; 335</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> M. Enock,   J.-M. Schwartz,   "Une dualité dans les algèbres de von Neumann" ''Bull. Soc. Math. France'' , '''44''' (1975) pp. 1–44</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J.-M. Schwartz,   "Relations entre "ring groups" et algèbres de Kac" ''Bull. Sci. Math. (2)'' , '''100''' (1976) pp. 289–300</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.G. Drinfel'd, "Quantum groups" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''1''' , Amer. Math. Soc. (1987) pp. 798–820</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.D. Faddeev, "Integrable models in (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310152.png" />)-dimensional quantum field theory" , ''Lectures in Les Houches, 1982'' , '''Session 39''' , Elsevier (1984) pp. 563–608 {{MR|782509}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Jimbo, "Quantum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310153.png" />-matrix for the generalized Toda system" ''Comm. Math. Phys.'' , '''102''' (1986) pp. 537–547 {{MR|824090}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Jimbo, "A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310154.png" />-difference analogue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310155.png" /> and the Yang–Baxter equation" ''Letters Math. Phys.'' , '''10''' (1985) pp. 63–69 {{MR|797001}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> V.G. Drinfel'd, "Hopf algebras and the quantum Yang–Baxter equation" ''Soviet Math. Dokl.'' , '''32''' (1985) pp. 254–258 ''Dokl. Akad. Nauk SSSR'' , '''283''' : 5 (1985) pp. 1060–1064</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> V.V. Lyubashenko, "Hopf algebras and vector symmetries" ''Russian Math. Surveys'' , '''41''' : 5 (1986) pp. 153–154 ''Uspekhi Mat. Nauk'' , '''41''' : 5 (1986) pp. 185–186 {{MR|0878344}} {{ZBL|0649.16008}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtayan, "Quantization of Lie groups and Lie algebras" ''Algebra and Analysis'' , '''1''' : 1 (1989) pp. 178–206 (In Russian) {{MR|1015339}} {{ZBL|0677.17010}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> N.Yu. Reshetikhin, "Quasitriangular Hopf algebras and invariants of tangles" ''Algebra and Analysis'' , '''1''' : 2 (1989) pp. 169–188 (In Russian) {{MR|}} {{ZBL|0715.17016}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> V.F.R. Jones, "A polynomial invariant for knots via von Neumann algebras" ''Bulletin Amer. Math. Soc.'' , '''12''' (1985) pp. 103–112 {{MR|0766964}} {{ZBL|0564.57006}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S.L. Woronowich, "Compact matrix pseudogroups" ''Comm. Math. Phys.'' , '''111''' (1987) pp. 613–665 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S.L. Woronowich, "Twisted SU(2) group. An example of a noncommutative differential calculus" ''Publ. RIMS'' , '''23''' (1987) pp. 117–181</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L.L. Vaksman, Ya.S. Soibelman, "Function algebra on the quantum group SU(2)" ''Funct. Anal. Appl.'' , '''22''' : 3 (1988) pp. 170–181 ''Funksional. Anal. Prilozhen.'' , '''22''' : 3 (1988) pp. 1–14</TD></TR><TR><TD valign="top">[a13a]</TD> <TD valign="top"> G.I. Kac, "Ring groups and the duality principle I" ''Trans. Moscow Math. Soc.'' , '''12''' (1963) pp. 291–339 ''Trudy Moskov. Mat. Obshch.'' , '''12''' (1963) pp. 295–301 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a13b]</TD> <TD valign="top"> G.I. Kac, "Ring groups and the duality principle II" ''Trans. Moscow Math. Soc.'' , '''13''' (1965) pp. 94–126 ''Trudy Moskov. Mat. Obshch.'' , '''13''' (1965) pp. 84–113 {{MR|}} {{ZBL|0162.45101}} {{ZBL|0144.37903}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> G.I. Kac, L.I. Vainerman, "Nonunimodular ring groups and Hopf–von Neumann algebras" ''Math. USSR Sb.'' , '''23''' (1974) pp. 185–214 ''Mat. Sb.'' , '''94''' : 2 (1974) pp. 194–225; 335 {{MR|0348038}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> M. Enock, J.-M. Schwartz, "Une dualité dans les algèbres de von Neumann" ''Bull. Soc. Math. France'' , '''44''' (1975) pp. 1–44 {{MR|0442710}} {{ZBL|0343.46044}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J.-M. Schwartz, "Relations entre "ring groups" et algèbres de Kac" ''Bull. Sci. Math. (2)'' , '''100''' (1976) pp. 289–300 {{MR|0473094}} {{ZBL|0343.46043}} </TD></TR></table>

Revision as of 14:51, 24 March 2012

The phrase "quantum group" is more or less a synonym for "Hopf algebra" . More precisely, the category of quantum groups is defined in [a1] to be dual to the category of Hopf algebras. This is natural for the following reason. There is the following general principle: The functor is an anti-equivalence between the category of "spaces" and the category of commutative associative unital algebras, perhaps with some additional structures or properties (this principle becomes a theorem if "space" is understood to be "affine scheme" or "compact topological space" , and "algebra" is understood to mean "C*-algebra" ). So one can translate the definition of a group into the language of algebras: instead of a space with an associative operation one obtains a commutative algebra over a commutative ring with a homomorphism , called comultiplication; the unit gives rise to a homomorphism , called co-unit, and the mapping , , gives rise to a bijective -linear mapping , called antipode. The group axioms are equivalent to the commutativity of the following diagrams:

Here , . The commutativity of these diagrams means that is a commutative Hopf algebra. Since the category of groups is anti-equivalent to the category of commutative Hopf algebras, it is natural to define a quantum group as an object of the category dual to the category of (not necessarily commutative) Hopf algebras.

A simple class of non-commutative Hopf algebras is formed by the group algebras of non-commutative groups. These Hopf algebras are commutative, i.e. is contained in the symmetric part of . Essentially, all cocommutative Hopf algebras are group algebras.

Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix and , where is a commutative ring. Denote by the associative -algebra with generators , , and defining relations if , if , if , , if , , , where is the number of inversions in the permutation . Then has a Hopf algebra structure defined by . If , then is the algebra of polynomial functions on . So, in the general case it is natural to consider elements of as "functions on the quantized SLn" .

The quantized is one of the simplest quantum groups which appear naturally in the theory of quantum integrable systems and, especially, in the quantum inverse-scattering method [a2]. The development of this method has led to the following quantization technique for constructing non-commutative non-cocommutative Hopf algebras. It is natural to construct them as deformations of commutative Hopf algebras. If a non-commutative deformation of a commutative Hopf algebra is given, then a Poisson bracket on is defined by , where is the deformation parameter and means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, ) and is compatible with comultiplication. In other words, is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra and then try to quantize it, i.e. to construct a Hopf algebra deformation of which induces the given Poisson bracket on .

Technically it is more convenient to deform not commutative Hopf algebras but cocommutative ones and to start not with a Poisson–Hopf algebra (or a Poisson–Lie group [a1], which is more or less the same) but with its infinitesimal version, called a Lie bi-algebra . A Lie bi-algebra is a Lie algebra with a linear mapping such that: 1) defines a Lie algebra structure on ; and 2) is a -cocycle ( acts on by means of the adjoint representation). By definition, a quantization of is a Hopf algebra deformation of the universal enveloping algebra such that , where is the Poisson cobracket, defined by . Here is the deformation parameter, the deformed comultiplication and the opposite comultiplication.

It is not known whether every Lie bi-algebra can be quantized, and usually quantization is not unique. But in several important cases (cf. [a1], §3, §6) there exists a canonical quantization. In particular, on a Kac–Moody algebra with a fixed scalar product there is a canonical Lie bi-algebra structure and this bi-algebra has a canonical quantization , as was discovered in [a3], [a4], [a5]. Let be the Cartan subalgebra of , the images of the simple roots . Then is generated by and , with the following defining relations:

Setting , one has also

Here is the Cartan matrix and is the Gauss polynomial, i.e.,

The comultiplication in is such that for and

If is a finite-dimensional simple Lie algebra (cf. Lie algebra, semi-simple), then the algebra of regular functions on the corresponding simply-connected algebraic group is isomorphic to the subalgebra of generated by the matrix elements of the finite-dimensional representations of . Therefore the subalgebra of generated by the matrix elements of the finite-dimensional representations of can be considered as the algebra of functions on a certain quantization of . For instance, the quantized (cf. above) can be obtained in this way.

There is an important notion of a quasitriangular Hopf algebra. This is a pair where is a Hopf algebra and is an invertible element of such that , , for . Here is the opposite comultiplication and , , are defined as follows: If , where , then , , . If is a quasitriangular Hopf algebra, then satisfies the quantum Yang–Baxter equation (cf. also Yang–Baxter equation), i.e., . It is known (cf. [a1], §13) that if is a finite-dimensional simple Lie algebra, then has a canonical quasitriangular structure, while if is an infinite-dimensional Kac–Moody algebra, then has an "almost quasitriangular" structure.

If is a quasitriangular Hopf algebra over and is a representation , then satisfies the quantum Yang–Baxter equation. There is an inverse construction (cf. [a6], [a7]), which goes back to the quantum inverse-scattering method: to a matrix solution of the quantum Yang–Baxter equation satisfying a non-degeneracy condition there corresponds a Hopf algebra. Without this condition one can only construct an associative bi-algebra (the difference between a Hopf algebra and an associative bi-algebra is that in the second case there may be no antipode). This bi-algebra is generated by elements , , with defining relations , where , , is the matrix , and is defined by .

Quasitriangular Hopf algebras are a natural tool for the quantum inverse-scattering in method ([a1], §11). On the other hand, they can be used (cf. [a8]) to construct invariants of knots (and of more general objects such as links and tangles) generalizing the Jones polynomial [a9]. More precisely, to an oriented knot and a quasitriangular Hopf algebra there corresponds a central element .

The usual notion of a group has several versions: abstract group, Lie group, topological group, etc. The same is true for quantum groups. The quantum analogue of the notion of a compact group was introduced in [a10] (the idea is to use -algebras instead of abstract algebras). The quantized (cf. [a11], [a12]) is a typical example. The notion of a ring group (cf. , [a14]) and the equivalent notion of a Kac algebra (cf. [a15], [a16]) were introduced as an attempt to define a locally compact quantum group. However, these notions are not general enough (the axioms of , [a14], [a15] imply that the square of the antipode is the identity mapping, and therefore the quantized is not a ring group).

References

[a1] V.G. Drinfel'd, "Quantum groups" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , 1 , Amer. Math. Soc. (1987) pp. 798–820
[a2] L.D. Faddeev, "Integrable models in ()-dimensional quantum field theory" , Lectures in Les Houches, 1982 , Session 39 , Elsevier (1984) pp. 563–608 MR782509
[a3] M. Jimbo, "Quantum -matrix for the generalized Toda system" Comm. Math. Phys. , 102 (1986) pp. 537–547 MR824090
[a4] M. Jimbo, "A -difference analogue of and the Yang–Baxter equation" Letters Math. Phys. , 10 (1985) pp. 63–69 MR797001
[a5] V.G. Drinfel'd, "Hopf algebras and the quantum Yang–Baxter equation" Soviet Math. Dokl. , 32 (1985) pp. 254–258 Dokl. Akad. Nauk SSSR , 283 : 5 (1985) pp. 1060–1064
[a6] V.V. Lyubashenko, "Hopf algebras and vector symmetries" Russian Math. Surveys , 41 : 5 (1986) pp. 153–154 Uspekhi Mat. Nauk , 41 : 5 (1986) pp. 185–186 MR0878344 Zbl 0649.16008
[a7] L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtayan, "Quantization of Lie groups and Lie algebras" Algebra and Analysis , 1 : 1 (1989) pp. 178–206 (In Russian) MR1015339 Zbl 0677.17010
[a8] N.Yu. Reshetikhin, "Quasitriangular Hopf algebras and invariants of tangles" Algebra and Analysis , 1 : 2 (1989) pp. 169–188 (In Russian) Zbl 0715.17016
[a9] V.F.R. Jones, "A polynomial invariant for knots via von Neumann algebras" Bulletin Amer. Math. Soc. , 12 (1985) pp. 103–112 MR0766964 Zbl 0564.57006
[a10] S.L. Woronowich, "Compact matrix pseudogroups" Comm. Math. Phys. , 111 (1987) pp. 613–665
[a11] S.L. Woronowich, "Twisted SU(2) group. An example of a noncommutative differential calculus" Publ. RIMS , 23 (1987) pp. 117–181
[a12] L.L. Vaksman, Ya.S. Soibelman, "Function algebra on the quantum group SU(2)" Funct. Anal. Appl. , 22 : 3 (1988) pp. 170–181 Funksional. Anal. Prilozhen. , 22 : 3 (1988) pp. 1–14
[a13a] G.I. Kac, "Ring groups and the duality principle I" Trans. Moscow Math. Soc. , 12 (1963) pp. 291–339 Trudy Moskov. Mat. Obshch. , 12 (1963) pp. 295–301
[a13b] G.I. Kac, "Ring groups and the duality principle II" Trans. Moscow Math. Soc. , 13 (1965) pp. 94–126 Trudy Moskov. Mat. Obshch. , 13 (1965) pp. 84–113 Zbl 0162.45101 Zbl 0144.37903
[a14] G.I. Kac, L.I. Vainerman, "Nonunimodular ring groups and Hopf–von Neumann algebras" Math. USSR Sb. , 23 (1974) pp. 185–214 Mat. Sb. , 94 : 2 (1974) pp. 194–225; 335 MR0348038
[a15] M. Enock, J.-M. Schwartz, "Une dualité dans les algèbres de von Neumann" Bull. Soc. Math. France , 44 (1975) pp. 1–44 MR0442710 Zbl 0343.46044
[a16] J.-M. Schwartz, "Relations entre "ring groups" et algèbres de Kac" Bull. Sci. Math. (2) , 100 (1976) pp. 289–300 MR0473094 Zbl 0343.46043
How to Cite This Entry:
Quantum groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_groups&oldid=21911
This article was adapted from an original article by V.G. Drinfel'd (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article