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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:
+
{{TEX|done}}
 +
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 $  X \mapsto \{ \textrm{ the  algebra  of  functions  on }  X \} $
 +
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 $  G $
 +
with an associative operation $  G \times G \rightarrow G $
 +
one obtains a commutative algebra $  A $
 +
over a commutative ring $  k $
 +
with a homomorphism $  \Delta : \  A \rightarrow A \otimes A $ ,  
 +
called comultiplication; the unit $  e \in G $
 +
gives rise to a homomorphism $  \epsilon : \  A \rightarrow k $ ,  
 +
called co-unit, and the mapping $  g \mapsto g ^{-1} $ ,  
 +
$  g \in G $ ,  
 +
gives rise to a bijective $  k $ -
 +
linear mapping $  S : \  A \rightarrow A \otimes A $ ,  
 +
called antipode. The group axioms are equivalent to the commutativity of the following diagrams:
 +
$$
 +
\begin{array}{crcrl}
 +
    &                    &  {A \otimes A}  &    & \\               
 +
{}  &\nearrow ^ \Delta    &    &\searrow ^ { \mathop{\rm id} \otimes \Delta}  &{}  \\
 +
A  &{}                &{}                  &{}      &  {A \otimes A \otimes A}  \\
 +
{}  &\searrow _ \Delta    &                &\nearrow _ {\Delta \otimes  \mathop{\rm id}}  &{}  \\
 +
    &                  &  {A \otimes A}  &      & \\
 +
\end{array}
 +
$$
  
<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/q07631014.png" /></td> </tr></table>
+
$$
 +
\begin{array}{ccc}
 +
A  &  \stackrel{ { \mathop{\rm id}\nolimits}} \rightarrow      & A  \\
 +
{\scriptsize \Delta} \downarrow    &{}  &\|  \\
 +
{A \otimes A}  &  \stackrel { \mathop{\rm id}\otimes\epsilon}\rightarrow    &{A \otimes k}  \\
 +
\end{array}
 +
\qquad\qquad   
 +
\begin{array}{ccc}
 +
A  &  \stackrel{ { \mathop{\rm id}\nolimits}} \rightarrow      & A  \\
 +
{\scriptsize \Delta} \downarrow    &{}  &\|  \\
 +
{A \otimes A}  &  \stackrel {\epsilon \otimes \mathop{\rm id} } \rightarrow  &{k \otimes A}  \\
 +
\end{array}
 +
$$
  
<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/q07631015.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/q07631016.png" /></td> </tr></table>
+
$$
 +
\begin{array}{ccccc}
 +
A  &  \stackrel \Delta  \rightarrow  A \otimes A  &\stackrel{ { \mathop{\rm id} \otimes S}} \rightarrow &A \otimes A \stackrel{m}\rightarrow    & A  \\
 +
{}  &\searrow _ \epsilon  &{}  &\nearrow _{i}  &{}  \\
 +
{}  &{}  & k  &{}  &{}  \\
 +
\end{array}
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631018.png" />. The commutativity of these diagrams means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631019.png" /> 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.
+
$$
 +
\begin{array}{ccccc}
 +
A  &  \stackrel \Delta  \rightarrow A \otimes A  & \stackrel{ {S \otimes \mathop{\rm id}}} \rightarrow  &A \otimes A
 +
\stackrel{m}\rightarrow  & A  \\
 +
{}  &\searrow _ \epsilon  &{}  &\nearrow _{i}  &{}  \\
 +
{}  &{}  & k  &{}  &{}  \\
 +
\end{array}
 +
$$
  
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 $  m(a \otimes b) = ab $ ,
 +
$  i(c) = c \cdot 1 _{A} $ .
 +
The commutativity of these diagrams means that $  (A ,\  \Delta ,\  \epsilon ,\  S ) $
 +
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.
  
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" .
+
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. $  \Delta (A) $
 +
is contained in the symmetric part of $  A \otimes A $ .  
 +
Essentially, all cocommutative Hopf algebras are group algebras.
  
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" />.
+
Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix $  n \in \mathbf N $
 +
and $  q \in k $ ,  
 +
where $  k $
 +
is a commutative ring. Denote by $  A $
 +
the associative $  k $ -
 +
algebra with generators $  x _{ij} $ ,
 +
$  1 \leq i,\  j \leq n $ ,
 +
and defining relations $  x _{ij} x _{il} = q x _{il} x _{ij} $
 +
if $  j < l $ ,
 +
$  x _{ij} x _{kj} = q x _{kj} x _{ij} $
 +
if $  i < k $ ,  
 +
$  x _{il} x _{kj} = x _{kj} x _{il} $
 +
if $  i < k $ ,
 +
$  l > j $ ,  
 +
$  [ x _{il} ,\  x _{kj} ] = (q ^{-1} -q) x _{ij} x _{kl} $
 +
if $  i > k $ ,
 +
$  l > j $ ,  
 +
$  \sum _ {i _{1} \dots i _{n}} x _ {1i _{1}} \dots x _ {ni _{n}} \cdot (-q) ^ {l(i _{1} \dots i _{n} )} = 1 $ ,  
 +
where $  l ( i _{1} \dots i _{n} ) $
 +
is the number of inversions in the permutation $  ( i _{1} \dots i _{n} ) $ .  
 +
Then $  A $
 +
has a Hopf algebra structure defined by $  \Delta (x _{ij} ) = \sum _{k} x _{ik} \otimes x _{kj} $ .  
 +
If $  q =1 $ ,
 +
then $  A $
 +
is the algebra of polynomial functions on $  \mathop{\rm SL}\nolimits (n) $ .  
 +
So, in the general case it is natural to consider elements of $  A $
 +
as "functions on the quantized SLn" .
  
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 [[#References|[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|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631060.png" /> with a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631061.png" /> such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png" /> defines a Lie algebra structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631063.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631064.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631065.png" />-cocycle (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631066.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631067.png" /> by means of the adjoint representation). By definition, a quantization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631068.png" /> is a Hopf algebra deformation of the [[Universal enveloping algebra|universal enveloping algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png" /> is the Poisson cobracket, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631073.png" /> is the deformation parameter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631074.png" /> the deformed comultiplication and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631075.png" /> the opposite comultiplication.
+
The quantized $  \mathop{\rm SL}\nolimits (n) $
 +
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 $  A $
 +
of a commutative Hopf algebra $  A _{0} $
 +
is given, then a Poisson bracket on $  A _{0} $
 +
is defined by $  \{ a,\  b \} = \mathop{\rm lim}\nolimits _ {h \rightarrow 0} \  h ^{-1} (ab-ba) $ ,
 +
where $  h $
 +
is the deformation parameter and $  ab $
 +
means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, $  \{ a,\  bc \} = \{ a,\  b \} c + \{ a,\  c \} b $ )
 +
and is compatible with comultiplication. In other words, $  A _{0} $
 +
is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra $  A _{0} $
 +
and then try to quantize it, i.e. to construct a Hopf algebra deformation of $  A _{0} $
 +
which induces the given Poisson bracket on $  A _{0} $ .
  
It is not known whether every Lie bi-algebra can be quantized, and usually quantization is not unique. But in several important cases (cf. [[#References|[a1]]], §3, §6) there exists a canonical quantization. In particular, on a [[Kac–Moody algebra|Kac–Moody algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631076.png" /> with a fixed scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631077.png" /> there is a canonical Lie bi-algebra structure and this bi-algebra has a canonical quantization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631078.png" />, as was discovered in [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631079.png" /> be the Cartan subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631081.png" /> the images of the simple roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631082.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631083.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631086.png" /> with the following defining relations:
 
  
<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/q07631087.png" /></td> </tr></table>
+
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 [[#References|[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|Lie algebra]] $  \mathfrak g $
 +
with a linear mapping $  \phi : \  \mathfrak g \rightarrow \mathfrak g \otimes \mathfrak g $
 +
such that: 1) $  \phi ^{*} : \  \mathfrak g ^{*} \otimes \mathfrak g ^{*} \rightarrow \mathfrak g ^{*} $
 +
defines a Lie algebra structure on $  \mathfrak g ^{*} $ ;  
 +
and 2) $  \phi $
 +
is a $  1 $ -
 +
cocycle ($  \mathfrak g $
 +
acts on $  \mathfrak g \otimes \mathfrak g $
 +
by means of the adjoint representation). By definition, a quantization of $  ( \mathfrak g ,\  \phi ) $
 +
is a Hopf algebra deformation of the [[Universal enveloping algebra|universal enveloping algebra]] $  U \mathfrak g $
 +
such that $  \delta \mid _ {\mathfrak g} = \phi $ ,
 +
where $  \delta : \  U \mathfrak g \rightarrow U \mathfrak g \otimes U \mathfrak g $
 +
is the Poisson cobracket, defined by $  \delta (a) = \mathop{\rm lim}\nolimits _ {h \rightarrow 0} \  h ^{-1} ( \Delta (a) - \Delta ^ \prime  (a)) $ .  
 +
Here $  h $
 +
is the deformation parameter, $  \Delta $
 +
the deformed comultiplication and $  \Delta ^ \prime  $
 +
the opposite comultiplication.
  
<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/q07631088.png" /></td> </tr></table>
+
It is not known whether every Lie bi-algebra can be quantized, and usually quantization is not unique. But in several important cases (cf. [[#References|[a1]]], §3, §6) there exists a canonical quantization. In particular, on a [[Kac–Moody algebra|Kac–Moody algebra]] $  \mathfrak g $
 +
with a fixed scalar product $  ( \  ,\  ) $
 +
there is a canonical Lie bi-algebra structure and this bi-algebra has a canonical quantization $  U _{h} \mathfrak g $ ,
 +
as was discovered in [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]. Let $  \mathfrak h $
 +
be the Cartan subalgebra of $  \mathfrak g $ ,
 +
$  H _{i} \in \mathfrak g $
 +
the images of the simple roots $  \alpha _{i} \in \mathfrak h ^{*} $ .
 +
Then $  U _{h} \mathfrak g $
 +
is generated by $  \mathfrak h $
 +
and $  X _{i} ^{+} $ ,
 +
$  X _{i} ^{-} $
 +
with the following defining relations:$$
 +
[a _{1} ,\  a _{2} ]  =   0    \textrm{ for }
 +
a _{1} ,\  a _{2} \in \mathfrak h ;
 +
$$
 +
$$
 +
[a ,\  X _{i} ^ \pm  ]  =   \pm \alpha _{i} (a
 +
) X _{i} ^ \pm    \textrm{ for }  a \in \mathfrak h ;
 +
$$
 +
$$
 +
[ X _{i} ^{+} ,\  X _{j} ^{-} ]  =   2 \delta
 +
_{ij} h ^{-1} \  \mathop{\rm sinh}\nolimits ( h H _{i} / 2) .
 +
$$
 +
Setting $  n = 1-A _{ij} $ ,
 +
$  q = \mathop{\rm exp}\nolimits \  h (H _{i} ,\  H _{j} )/2 $
 +
one has also$$
 +
\sum _{k=0} ^ n (-1) ^{k} \binom{n}{k} _{q}
 +
q ^ {-k(n-k)/2} (X _{i} ^ \pm  ) ^{k} \cdot
 +
X _{j} ^ \pm  \cdot (X _{i} ^ \pm  ) ^{n-k}  =   0 .
 +
$$
 +
Here $  (A _{ij} ) $
 +
is the [[Cartan matrix|Cartan matrix]] and $  ( {} _{k} ^{n} ) _{q} $
 +
is the Gauss polynomial, i.e.,$$
 +
\binom{n}{k} _{q}  = 
  
<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/q07631089.png" /></td> </tr></table>
+
\frac{(q ^{n} -1 ) \dots (q ^{n-k+1} -1)}{( q ^{k} -1 ) \dots (q-1)}
 +
.
 +
$$
 +
The comultiplication in $  U _{h} \mathfrak g $
 +
is such that $  \Delta (a) = a \otimes 1 + 1 \otimes a $
 +
for $  a \in \mathfrak h $
 +
and$$
 +
\Delta (X _{i} ^ \pm  )  =
 +
X _{i} ^ \pm  \otimes  \mathop{\rm exp}\nolimits \left (
 +
\frac{hH _{i}}{4}
 +
\right ) +
 +
\mathop{\rm exp}\nolimits \left (
 +
\frac{-hH _{i}}{4}
 +
\right ) \otimes X _{i} ^ \pm  .
 +
$$
 +
If $  \mathfrak g $
 +
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 $  G $
 +
is isomorphic to the subalgebra of $  ( U \mathfrak g ) ^{*} $
 +
generated by the matrix elements of the finite-dimensional representations of $  U \mathfrak g $ .
 +
Therefore the subalgebra of $  ( U _{h} \mathfrak g ) ^{*} $
 +
generated by the matrix elements of the finite-dimensional representations of $  U \mathfrak g $
 +
can be considered as the algebra of functions on a certain quantization of $  G $ .  
 +
For instance, the quantized $  \mathop{\rm SL}\nolimits ( n ) $ (
 +
cf. above) can be obtained in this way.
  
Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631091.png" /> one has also
+
There is an important notion of a quasitriangular Hopf algebra. This is a pair $  (A,\  R) $
 +
where $  A $
 +
is a Hopf algebra and $  R $
 +
is an invertible element of $  A \otimes A $
 +
such that $  ( \Delta \otimes  \mathop{\rm id}\nolimits ) (R) = R ^{13} R ^{23} $ ,
 +
$  (  \mathop{\rm id}\nolimits \otimes \Delta )(R) = R ^{13} R ^{12} $ ,
 +
$  \Delta ^ \prime  (a) = R \cdot \Delta ( a ) \cdot R ^{-1} $
 +
for $  a \in A $ .
 +
Here $  \Delta ^ \prime  $
 +
is the opposite comultiplication and $  R ^{12} $ ,
 +
$  R ^{13} $ ,
 +
$  R ^{23} $
 +
are defined as follows: If $  R = \sum _{i} x _{i} \otimes y _{i} $ ,
 +
where $  x _{i} ,\  y _{i} \in A $ ,
 +
then $  R ^{12} = \sum _{i} x _{i} \otimes y _{i} \otimes 1 $ ,
 +
$  R ^{13} = \sum _{i} x _{i} \otimes 1 \otimes y _{i} $ ,
 +
$  R ^{23} = \sum _{i} 1 \otimes x _{i} \otimes y _{i} $ .
 +
If $  (A,\  R ) $
 +
is a quasitriangular Hopf algebra, then $  R $
 +
satisfies the quantum Yang–Baxter equation (cf. also [[Yang–Baxter equation|Yang–Baxter equation]]), i.e., $  R ^{12} R ^{13} R ^{23} = R ^{23} R ^{13} R ^{12} $ .
 +
It is known (cf. [[#References|[a1]]], §13) that if $  \mathfrak g $
 +
is a finite-dimensional simple Lie algebra, then $  U _{h} \mathfrak g $
 +
has a canonical quasitriangular structure, while if $  \mathfrak g $
 +
is an infinite-dimensional Kac–Moody algebra, then $  U _{h} \mathfrak g $
 +
has an "almost quasitriangular" structure.
  
<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/q07631092.png" /></td> </tr></table>
+
If $  (A,\  R) $
 +
is a quasitriangular Hopf algebra over $  k $
 +
and $  \rho $
 +
is a representation $  A \rightarrow  \mathop{\rm Mat}\nolimits (n,\  k) $ ,
 +
then $  {\mathcal R} = ( \rho \otimes \rho )(R) \in  \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} ) $
 +
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 $  t _{ij} $ ,
 +
$  1 \leq i,\  j \leq n $ ,
 +
with defining relations $  {\mathcal R} T _{1} T _{2} = T _{2} T _{1} {\mathcal R} $ ,
 +
where $  T _{1} = T \otimes 1 \in  \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} ) $ ,
 +
$  T _{2} = 1 \otimes T \in  \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} ) $ ,
 +
$  T $
 +
is the matrix $  (t _{ij} ) $ ,
 +
and $  \Delta $
 +
is defined by $  \Delta (t _{ij} ) = \sum _{k} t _{ik} \otimes t _{kj} $ .
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631093.png" /> is the [[Cartan matrix|Cartan matrix]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631094.png" /> is the Gauss polynomial, i.e.,
 
  
<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/q07631095.png" /></td> </tr></table>
+
Quasitriangular Hopf algebras are a natural tool for the quantum inverse-scattering in method ([[#References|[a1]]], §11). On the other hand, they can be used (cf. [[#References|[a8]]]) to construct invariants of knots (and of more general objects such as links and tangles) generalizing the Jones polynomial [[#References|[a9]]]. More precisely, to an oriented knot $  \gamma \subset \mathbf R ^{3} $
 +
and a quasitriangular Hopf algebra $  (A ,\  R) $
 +
there corresponds a central element $  z _ \gamma  \in A $ .
  
The comultiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631096.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631097.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631098.png" /> and
 
  
<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/q07631099.png" /></td> </tr></table>
+
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 [[#References|[a10]]] (the idea is to use $  C ^{*} $ -
 
+
algebras instead of abstract algebras). The quantized $  \mathop{\rm SU}\nolimits (2) $ (
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.
+
cf. [[#References|[a11]]], [[#References|[a12]]]) is a typical example. The notion of a ring group (cf. , [[#References|[a14]]]) and the equivalent notion of a Kac algebra (cf. [[#References|[a15]]], [[#References|[a16]]]) were introduced as an attempt to define a locally compact quantum group. However, these notions are not general enough (the axioms of , [[#References|[a14]]], [[#References|[a15]]] imply that the square of the antipode is the identity mapping, and therefore the quantized $  \mathop{\rm SU}\nolimits (2) $
 
+
is not a ring group).
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" />.
 
 
 
Quasitriangular Hopf algebras are a natural tool for the quantum inverse-scattering in method ([[#References|[a1]]], §11). On the other hand, they can be used (cf. [[#References|[a8]]]) to construct invariants of knots (and of more general objects such as links and tangles) generalizing the Jones polynomial [[#References|[a9]]]. More precisely, to an oriented knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310146.png" /> and a quasitriangular Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310147.png" /> there corresponds a central element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310148.png" />.
 
 
 
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 [[#References|[a10]]] (the idea is to use <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310149.png" />-algebras instead of abstract algebras). The quantized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310150.png" /> (cf. [[#References|[a11]]], [[#References|[a12]]]) is a typical example. The notion of a ring group (cf. , [[#References|[a14]]]) and the equivalent notion of a Kac algebra (cf. [[#References|[a15]]], [[#References|[a16]]]) were introduced as an attempt to define a locally compact quantum group. However, these notions are not general enough (the axioms of , [[#References|[a14]]], [[#References|[a15]]] imply that the square of the antipode is the identity mapping, and therefore the quantized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q076310151.png" /> is not a ring group).
 
  
 
====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 {{MR|}} {{ZBL|0667.16003}} </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 {{MR|}} {{ZBL|0588.17015}} </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>
 
<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 {{MR|}} {{ZBL|0667.16003}} </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 {{MR|}} {{ZBL|0588.17015}} </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>

Latest revision as of 19:09, 16 December 2019

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 $ X \mapsto \{ \textrm{ the algebra of functions on } X \} $ 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 $ G $ with an associative operation $ G \times G \rightarrow G $ one obtains a commutative algebra $ A $ over a commutative ring $ k $ with a homomorphism $ \Delta : \ A \rightarrow A \otimes A $ , called comultiplication; the unit $ e \in G $ gives rise to a homomorphism $ \epsilon : \ A \rightarrow k $ , called co-unit, and the mapping $ g \mapsto g ^{-1} $ , $ g \in G $ , gives rise to a bijective $ k $ - linear mapping $ S : \ A \rightarrow A \otimes A $ , called antipode. The group axioms are equivalent to the commutativity of the following diagrams: $$ \begin{array}{crcrl} & & {A \otimes A} & & \\ {} &\nearrow ^ \Delta & &\searrow ^ { \mathop{\rm id} \otimes \Delta} &{} \\ A &{} &{} &{} & {A \otimes A \otimes A} \\ {} &\searrow _ \Delta & &\nearrow _ {\Delta \otimes \mathop{\rm id}} &{} \\ & & {A \otimes A} & & \\ \end{array} $$


$$ \begin{array}{ccc} A & \stackrel{ { \mathop{\rm id}\nolimits}} \rightarrow & A \\ {\scriptsize \Delta} \downarrow &{} &\| \\ {A \otimes A} & \stackrel { \mathop{\rm id}\otimes\epsilon}\rightarrow &{A \otimes k} \\ \end{array} \qquad\qquad \begin{array}{ccc} A & \stackrel{ { \mathop{\rm id}\nolimits}} \rightarrow & A \\ {\scriptsize \Delta} \downarrow &{} &\| \\ {A \otimes A} & \stackrel {\epsilon \otimes \mathop{\rm id} } \rightarrow &{k \otimes A} \\ \end{array} $$


$$ \begin{array}{ccccc} A & \stackrel \Delta \rightarrow A \otimes A &\stackrel{ { \mathop{\rm id} \otimes S}} \rightarrow &A \otimes A \stackrel{m}\rightarrow & A \\ {} &\searrow _ \epsilon &{} &\nearrow _{i} &{} \\ {} &{} & k &{} &{} \\ \end{array} $$

$$ \begin{array}{ccccc} A & \stackrel \Delta \rightarrow A \otimes A & \stackrel{ {S \otimes \mathop{\rm id}}} \rightarrow &A \otimes A \stackrel{m}\rightarrow & A \\ {} &\searrow _ \epsilon &{} &\nearrow _{i} &{} \\ {} &{} & k &{} &{} \\ \end{array} $$

Here $ m(a \otimes b) = ab $ , $ i(c) = c \cdot 1 _{A} $ . The commutativity of these diagrams means that $ (A ,\ \Delta ,\ \epsilon ,\ S ) $ 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. $ \Delta (A) $ is contained in the symmetric part of $ A \otimes A $ . Essentially, all cocommutative Hopf algebras are group algebras.

Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix $ n \in \mathbf N $ and $ q \in k $ , where $ k $ is a commutative ring. Denote by $ A $ the associative $ k $ - algebra with generators $ x _{ij} $ , $ 1 \leq i,\ j \leq n $ , and defining relations $ x _{ij} x _{il} = q x _{il} x _{ij} $ if $ j < l $ , $ x _{ij} x _{kj} = q x _{kj} x _{ij} $ if $ i < k $ , $ x _{il} x _{kj} = x _{kj} x _{il} $ if $ i < k $ , $ l > j $ , $ [ x _{il} ,\ x _{kj} ] = (q ^{-1} -q) x _{ij} x _{kl} $ if $ i > k $ , $ l > j $ , $ \sum _ {i _{1} \dots i _{n}} x _ {1i _{1}} \dots x _ {ni _{n}} \cdot (-q) ^ {l(i _{1} \dots i _{n} )} = 1 $ , where $ l ( i _{1} \dots i _{n} ) $ is the number of inversions in the permutation $ ( i _{1} \dots i _{n} ) $ . Then $ A $ has a Hopf algebra structure defined by $ \Delta (x _{ij} ) = \sum _{k} x _{ik} \otimes x _{kj} $ . If $ q =1 $ , then $ A $ is the algebra of polynomial functions on $ \mathop{\rm SL}\nolimits (n) $ . So, in the general case it is natural to consider elements of $ A $ as "functions on the quantized SLn" .

The quantized $ \mathop{\rm SL}\nolimits (n) $ 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 $ A $ of a commutative Hopf algebra $ A _{0} $ is given, then a Poisson bracket on $ A _{0} $ is defined by $ \{ a,\ b \} = \mathop{\rm lim}\nolimits _ {h \rightarrow 0} \ h ^{-1} (ab-ba) $ , where $ h $ is the deformation parameter and $ ab $ means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, $ \{ a,\ bc \} = \{ a,\ b \} c + \{ a,\ c \} b $ ) and is compatible with comultiplication. In other words, $ A _{0} $ is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra $ A _{0} $ and then try to quantize it, i.e. to construct a Hopf algebra deformation of $ A _{0} $ which induces the given Poisson bracket on $ A _{0} $ .


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 $ \mathfrak g $ with a linear mapping $ \phi : \ \mathfrak g \rightarrow \mathfrak g \otimes \mathfrak g $ such that: 1) $ \phi ^{*} : \ \mathfrak g ^{*} \otimes \mathfrak g ^{*} \rightarrow \mathfrak g ^{*} $ defines a Lie algebra structure on $ \mathfrak g ^{*} $ ; and 2) $ \phi $ is a $ 1 $ - cocycle ($ \mathfrak g $ acts on $ \mathfrak g \otimes \mathfrak g $ by means of the adjoint representation). By definition, a quantization of $ ( \mathfrak g ,\ \phi ) $ is a Hopf algebra deformation of the universal enveloping algebra $ U \mathfrak g $ such that $ \delta \mid _ {\mathfrak g} = \phi $ , where $ \delta : \ U \mathfrak g \rightarrow U \mathfrak g \otimes U \mathfrak g $ is the Poisson cobracket, defined by $ \delta (a) = \mathop{\rm lim}\nolimits _ {h \rightarrow 0} \ h ^{-1} ( \Delta (a) - \Delta ^ \prime (a)) $ . Here $ h $ is the deformation parameter, $ \Delta $ the deformed comultiplication and $ \Delta ^ \prime $ 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 $ \mathfrak g $ with a fixed scalar product $ ( \ ,\ ) $ there is a canonical Lie bi-algebra structure and this bi-algebra has a canonical quantization $ U _{h} \mathfrak g $ , as was discovered in [a3], [a4], [a5]. Let $ \mathfrak h $ be the Cartan subalgebra of $ \mathfrak g $ , $ H _{i} \in \mathfrak g $ the images of the simple roots $ \alpha _{i} \in \mathfrak h ^{*} $ . Then $ U _{h} \mathfrak g $ is generated by $ \mathfrak h $ and $ X _{i} ^{+} $ , $ X _{i} ^{-} $ with the following defining relations:$$ [a _{1} ,\ a _{2} ] = 0 \textrm{ for } a _{1} ,\ a _{2} \in \mathfrak h ; $$ $$ [a ,\ X _{i} ^ \pm ] = \pm \alpha _{i} (a ) X _{i} ^ \pm \textrm{ for } a \in \mathfrak h ; $$ $$ [ X _{i} ^{+} ,\ X _{j} ^{-} ] = 2 \delta _{ij} h ^{-1} \ \mathop{\rm sinh}\nolimits ( h H _{i} / 2) . $$ Setting $ n = 1-A _{ij} $ , $ q = \mathop{\rm exp}\nolimits \ h (H _{i} ,\ H _{j} )/2 $ one has also$$ \sum _{k=0} ^ n (-1) ^{k} \binom{n}{k} _{q} q ^ {-k(n-k)/2} (X _{i} ^ \pm ) ^{k} \cdot X _{j} ^ \pm \cdot (X _{i} ^ \pm ) ^{n-k} = 0 . $$ Here $ (A _{ij} ) $ is the Cartan matrix and $ ( {} _{k} ^{n} ) _{q} $ is the Gauss polynomial, i.e.,$$ \binom{n}{k} _{q} = \frac{(q ^{n} -1 ) \dots (q ^{n-k+1} -1)}{( q ^{k} -1 ) \dots (q-1)} . $$ The comultiplication in $ U _{h} \mathfrak g $ is such that $ \Delta (a) = a \otimes 1 + 1 \otimes a $ for $ a \in \mathfrak h $ and$$ \Delta (X _{i} ^ \pm ) = X _{i} ^ \pm \otimes \mathop{\rm exp}\nolimits \left ( \frac{hH _{i}}{4} \right ) + \mathop{\rm exp}\nolimits \left ( \frac{-hH _{i}}{4} \right ) \otimes X _{i} ^ \pm . $$ If $ \mathfrak g $ 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 $ G $ is isomorphic to the subalgebra of $ ( U \mathfrak g ) ^{*} $ generated by the matrix elements of the finite-dimensional representations of $ U \mathfrak g $ . Therefore the subalgebra of $ ( U _{h} \mathfrak g ) ^{*} $ generated by the matrix elements of the finite-dimensional representations of $ U \mathfrak g $ can be considered as the algebra of functions on a certain quantization of $ G $ . For instance, the quantized $ \mathop{\rm SL}\nolimits ( n ) $ ( cf. above) can be obtained in this way.

There is an important notion of a quasitriangular Hopf algebra. This is a pair $ (A,\ R) $ where $ A $ is a Hopf algebra and $ R $ is an invertible element of $ A \otimes A $ such that $ ( \Delta \otimes \mathop{\rm id}\nolimits ) (R) = R ^{13} R ^{23} $ , $ ( \mathop{\rm id}\nolimits \otimes \Delta )(R) = R ^{13} R ^{12} $ , $ \Delta ^ \prime (a) = R \cdot \Delta ( a ) \cdot R ^{-1} $ for $ a \in A $ . Here $ \Delta ^ \prime $ is the opposite comultiplication and $ R ^{12} $ , $ R ^{13} $ , $ R ^{23} $ are defined as follows: If $ R = \sum _{i} x _{i} \otimes y _{i} $ , where $ x _{i} ,\ y _{i} \in A $ , then $ R ^{12} = \sum _{i} x _{i} \otimes y _{i} \otimes 1 $ , $ R ^{13} = \sum _{i} x _{i} \otimes 1 \otimes y _{i} $ , $ R ^{23} = \sum _{i} 1 \otimes x _{i} \otimes y _{i} $ . If $ (A,\ R ) $ is a quasitriangular Hopf algebra, then $ R $ satisfies the quantum Yang–Baxter equation (cf. also Yang–Baxter equation), i.e., $ R ^{12} R ^{13} R ^{23} = R ^{23} R ^{13} R ^{12} $ . It is known (cf. [a1], §13) that if $ \mathfrak g $ is a finite-dimensional simple Lie algebra, then $ U _{h} \mathfrak g $ has a canonical quasitriangular structure, while if $ \mathfrak g $ is an infinite-dimensional Kac–Moody algebra, then $ U _{h} \mathfrak g $ has an "almost quasitriangular" structure.

If $ (A,\ R) $ is a quasitriangular Hopf algebra over $ k $ and $ \rho $ is a representation $ A \rightarrow \mathop{\rm Mat}\nolimits (n,\ k) $ , then $ {\mathcal R} = ( \rho \otimes \rho )(R) \in \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} ) $ 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 $ t _{ij} $ , $ 1 \leq i,\ j \leq n $ , with defining relations $ {\mathcal R} T _{1} T _{2} = T _{2} T _{1} {\mathcal R} $ , where $ T _{1} = T \otimes 1 \in \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} ) $ , $ T _{2} = 1 \otimes T \in \mathop{\rm End}\nolimits (k ^{n} \otimes k ^{n} ) $ , $ T $ is the matrix $ (t _{ij} ) $ , and $ \Delta $ is defined by $ \Delta (t _{ij} ) = \sum _{k} t _{ik} \otimes t _{kj} $ .


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 $ \gamma \subset \mathbf R ^{3} $ and a quasitriangular Hopf algebra $ (A ,\ R) $ there corresponds a central element $ z _ \gamma \in A $ .


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 $ C ^{*} $ - algebras instead of abstract algebras). The quantized $ \mathop{\rm SU}\nolimits (2) $ ( 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 $ \mathop{\rm SU}\nolimits (2) $ 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 Zbl 0667.16003
[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 Zbl 0588.17015
[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=21992
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