View source for Quantum groups

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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 $  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 [[#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} $ .


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.

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}   =  

\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.

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.

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} $ .


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 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) $ (
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).

====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>