Editing Quantum group (section)

==Drinfeld–Jimbo type quantum groups==
One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the [[universal enveloping algebra]] of a [[semisimple Lie algebra]] or, more generally, a [[Kac–Moody algebra]], in the category of [[Hopf algebra]]s. The resulting algebra has additional structure, making it into a [[quasitriangular Hopf algebra]].

Let ''A'' = (''a<sub>ij</sub>'') be the [[Cartan matrix]] of the Kac–Moody algebra, and let ''q'' ≠ 0, 1 be a complex number, then the quantum group, ''U<sub>q</sub>''(''G''), where ''G'' is the Lie algebra whose Cartan matrix is ''A'', is defined as the [[unital algebra|unital]] [[associative algebra]] with generators ''k<sub>λ</sub>'' (where ''λ'' is an element of the [[weight lattice]], i.e. 2(λ, α<sub>''i''</sub>)/(α<sub>''i''</sub>, α<sub>''i''</sub>) is an integer for all ''i''), and ''e<sub>i</sub>'' and ''f<sub>i</sub>'' (for [[Root system#Positive roots and simple roots|simple root]]s, α<sub>''i''</sub>), subject to the following relations:

:<math>\begin{align}
k_0 &= 1 \\
k_\lambda k_\mu &= k_{\lambda+\mu} \\
k_\lambda e_i k_\lambda^{-1} &= q^{(\lambda,\alpha_i)} e_i \\
k_\lambda f_i k_\lambda^{-1} &= q^{- (\lambda,\alpha_i)} f_i \\
 \left [e_i, f_j \right ] &= \delta_{ij} \frac{k_i - k_i^{-1}}{q_i - q_i^{-1}} && k_i = k_{\alpha_i}, q_i = q^{\frac{1}{2}(\alpha_i,\alpha_i)} \\
\end{align}</math>

And for  ''i'' ≠ ''j'' we have the ''q''-Serre relations, which are deformations of the [[Jean-Pierre Serre|Serre]] relations:

:<math>\begin{align}
\sum_{n=0}^{1 - a_{ij}} (-1)^n \frac{[1 - a_{ij}]_{q_i}!}{[1 - a_{ij} - n]_{q_i}! [n]_{q_i}!} e_i^n e_j e_i^{1 - a_{ij} - n} &= 0 \\[6pt]
\sum_{n=0}^{1 - a_{ij}} (-1)^n \frac{[1 - a_{ij}]_{q_i}!}{[1 - a_{ij} - n]_{q_i}! [n]_{q_i}!} f_i^n f_j f_i^{1 - a_{ij} - n} &= 0
\end{align}</math>

where the [[q-factorial]], the [[q-analog]] of the ordinary [[factorial]], is defined recursively using q-number:

:<math>\begin{align} 
 {[0]}_{q_i}! &= 1 \\ 
{[n]}_{q_i}! &= \prod_{m=1}^n [m]_{q_i}, && [m]_{q_i} = \frac{q_i^m - q_i^{-m}}{q_i - q_i^{-1}}
\end{align}</math>

In the limit as ''q'' → 1, these relations approach the relations for the universal enveloping algebra ''U''(''G''), where

:<math>k_{\lambda} \to 1, \qquad \frac{k_\lambda - k_{-\lambda}}{q - q^{-1}} \to t_\lambda</math>

and ''t<sub>λ</sub>'' is the element of the Cartan subalgebra satisfying (''t<sub>λ</sub>'', ''h'') = ''λ''(''h'') for all ''h'' in the Cartan subalgebra.

There are various [[coalgebra|coassociative coproducts]] under which these algebras are Hopf algebras, for example,

:<math> \begin{array}{lll}
\Delta_1(k_\lambda) = k_\lambda \otimes k_\lambda  & \Delta_1(e_i) = 1 \otimes e_i + e_i \otimes k_i & \Delta_1(f_i) = k_i^{-1} \otimes f_i + f_i \otimes 1 \\
\Delta_2(k_\lambda) = k_\lambda \otimes k_\lambda & \Delta_2(e_i) = k_i^{-1} \otimes e_i + e_i \otimes 1 & \Delta_2(f_i) = 1 \otimes f_i + f_i \otimes k_i \\
\Delta_3(k_\lambda) = k_\lambda \otimes k_\lambda & \Delta_3(e_i) = k_i^{-\frac{1}{2}} \otimes e_i + e_i \otimes k_i^{\frac{1}{2}} & \Delta_3(f_i) = k_i^{-\frac{1}{2}} \otimes f_i + f_i \otimes k_i^{\frac{1}{2}}
\end{array}</math>

where the set of generators has been extended, if required, to include ''k<sub>λ</sub>'' for ''λ'' which is expressible as the sum of an element of the weight lattice and half an element of the [[root lattice]].

In addition, any Hopf algebra  leads to another with reversed coproduct ''T'' <small> o </small> Δ, where ''T'' is given by ''T''(''x'' ⊗ ''y'') = ''y'' ⊗ ''x'', giving three more possible versions.

The [[counit]] on ''U''<sub>''q''</sub>(''A'') is the same for all these coproducts: ''ε''(''k<sub>λ</sub>'') = 1, ''ε''(''e<sub>i</sub>'') = ''ε''(''f<sub>i</sub>'') = 0, and the respective [[Hopf algebra|antipodes]] for the above coproducts are given by

:<math> \begin{array}{lll}
S_1(k_\lambda) = k_{-\lambda} & S_1(e_i) = - e_i k_i^{-1} & S_1(f_i) = - k_i f_i \\
S_2(k_\lambda) = k_{-\lambda} & S_2(e_i) = - k_i e_i & S_2(f_i) = - f_i k_i^{-1} \\
S_3(k_\lambda) = k_{-\lambda} & S_3(e_i) = - q_i e_i & S_3(f_i) = - q_i^{-1} f_i
\end{array}</math>

Alternatively, the quantum group ''U''<sub>''q''</sub>(''G'') can be regarded as an algebra over the field '''C'''(''q''), the field of all [[rational function]]s of an indeterminate ''q'' over '''C'''.

Similarly, the quantum group ''U''<sub>''q''</sub>(''G'') can be regarded as an algebra over the field '''Q'''(''q''), the field of all [[rational function]]s of an indeterminate ''q'' over '''Q''' (see below in the section on quantum groups at ''q'' = 0). The center of quantum group can be described by quantum determinant.

===Representation theory===
Just as there are many different types of representations for Kac–Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups.

As is the case for all Hopf algebras, ''U<sub>q</sub>''(''G'') has an [[adjoint endomorphism|adjoint representation]] on itself as a module, with the action being given by 
:<math>\mathrm{Ad}_x \cdot y = \sum_{(x)} x_{(1)} y S(x_{(2)}),</math>
where 
:<math>\Delta(x) = \sum_{(x)} x_{(1)} \otimes x_{(2)}.</math>

====Case 1: ''q'' is not a root of unity====
One important type of representation is a weight representation, and the corresponding [[Module (mathematics)|module]] is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector ''v'' such that ''k<sub>λ</sub>'' · ''v'' = ''d<sub>λ</sub>v'' for all ''λ'', where ''d<sub>λ</sub>'' are complex numbers for all weights ''λ'' such that

:<math>d_0 = 1,</math>
:<math>d_\lambda d_\mu = d_{\lambda + \mu},</math> for all weights ''λ'' and ''μ''.

A weight module is called integrable if the actions of ''e<sub>i</sub>'' and ''f<sub>i</sub>''  are locally nilpotent (i.e. for any vector ''v'' in the module, there exists a positive integer ''k'', possibly dependent on ''v'',  such that <math>e_i^k.v = f_i^k.v = 0</math> for all ''i'').  In the case of integrable modules, the complex numbers ''d''<sub>''λ''</sub> associated with a weight vector satisfy <math>d_\lambda = c_\lambda q^{(\lambda,\nu)}</math>,{{Citation needed|date=July 2016}} where ''ν'' is an element of the weight lattice, and ''c<sub>λ</sub>'' are complex numbers such that

:*<math>c_0 = 1,</math>
:*<math>c_\lambda c_\mu = c_{\lambda + \mu},</math> for all weights ''λ'' and ''μ'',
:*<math>c_{2\alpha_i} = 1</math> for all ''i''.

Of special interest are [[highest-weight representation]]s, and the corresponding highest weight modules.  A highest weight module is a module generated by a weight vector ''v'', subject to ''k''<sub>''λ''</sub> · ''v'' = ''d<sub>λ</sub>v'' for all weights ''μ'', and ''e<sub>i</sub>'' · ''v'' = 0 for all ''i''.  Similarly, a quantum group can have a lowest weight representation and lowest weight module, ''i.e.'' a module generated by a weight vector ''v'', subject to ''k<sub>λ</sub>'' · ''v'' = ''d<sub>λ</sub>v'' for all weights ''λ'', and ''f<sub>i</sub>'' · ''v'' = 0 for all ''i''.

Define a vector ''v'' to have weight ''ν'' if <math>k_\lambda\cdot v = q^{(\lambda,\nu)} v</math> for all ''λ'' in the weight lattice.

If ''G'' is a Kac–Moody algebra, then in any irreducible highest weight representation of ''U''<sub>''q''</sub>(''G''), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation of ''U''(''G'') with equal highest weight.  If the highest weight is dominant and integral (a weight ''μ'' is dominant and integral if ''μ'' satisfies the condition that <math>2 (\mu,\alpha_i)/(\alpha_i,\alpha_i)</math> is a non-negative integer for all ''i''), then the weight spectrum of the irreducible representation is invariant under the [[Weyl group]] for ''G'', and the representation is integrable.

Conversely, if a highest weight module is integrable, then its highest weight vector ''v'' satisfies <math>k_\lambda\cdot v = c_\lambda q^{(\lambda,\nu)} v</math>, where ''c''<sub>''λ''</sub> · ''v'' = ''d''<sub>''λ''</sub>''v'' are complex numbers such that

:*<math>c_0 = 1,</math>
:*<math>c_\lambda c_\mu = c_{\lambda + \mu},</math> for all weights ''λ'' and ''μ'',
:*<math>c_{2\alpha_i} = 1</math> for all ''i'',

and ''ν'' is dominant and integral.

As is the case for all Hopf algebras, the [[tensor product]] of two modules is another module.  For an element ''x'' of ''U<sub>q</sub>(G)'', and for vectors ''v'' and ''w'' in the respective modules, ''x'' ⋅ (''v'' ⊗ ''w'') = Δ(''x'') ⋅ (''v'' ⊗ ''w''), so that <math>k_\lambda\cdot(v \otimes w) = k_\lambda\cdot v \otimes k_\lambda.w</math>, and in the case of coproduct Δ<sub>1</sub>, <math>e_i\cdot(v \otimes w) = k_i\cdot v \otimes e_i\cdot w + e_i\cdot v \otimes w</math> and <math>f_i\cdot(v \otimes w) = v \otimes f_i\cdot w + f_i\cdot v \otimes k_i^{-1}\cdot w.</math>

The integrable highest weight module described above is a tensor product of a one-dimensional module (on which ''k''<sub>λ</sub> = ''c''<sub>''λ''</sub> for all ''λ'', and ''e<sub>i</sub>'' = ''f<sub>i</sub>'' = 0 for all ''i'') and a highest weight module generated by a nonzero vector ''v''<sub>0</sub>, subject to <math>k_\lambda\cdot v_0 = q^{(\lambda,\nu)} v_0</math> for all weights ''λ'', and <math>e_i\cdot v_0 = 0</math> for all ''i''.

In the specific case where ''G'' is a finite-dimensional Lie algebra (as a special case of a Kac–Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional.

In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac–Moody algebra (the highest weights are the same, as are their multiplicities).

====Case 2: ''q'' is a root of unity====
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===Quasitriangularity===

====Case 1: ''q'' is not a root of unity====
Strictly, the quantum group ''U''<sub>''q''</sub>(''G'') is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an [[R-matrix|''R''-matrix]].  This infinite formal sum is expressible in terms of generators ''e<sub>i</sub>'' and ''f<sub>i</sub>'', and Cartan generators ''t''<sub>''λ''</sub>, where ''k<sub>λ</sub>'' is formally identified with ''q''<sup>''t''<sub>''λ''</sub></sup>.  The infinite formal sum is the product of two factors,{{citation needed|reason=I could not find this in references or anywhere else. Chari-Pressley has a different formula.|date=July 2016}}
:<math>q^{\eta \sum_j t_{\lambda_j} \otimes t_{\mu_j}}</math>
and an infinite formal sum, where ''λ''<sub>''j''</sub> is a basis for the dual space to the Cartan subalgebra, and ''μ''<sub>''j''</sub> is the dual basis, and ''η'' = ±1.

The formal infinite sum which plays the part of the [[R-matrix|''R''-matrix]] has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product of two lowest weight modules.  Specifically, if ''v'' has weight ''α'' and ''w'' has weight ''β'', then 
:<math>q^{\eta \sum_j t_{\lambda_j} \otimes t_{\mu_j}}\cdot(v \otimes w) = q^{\eta (\alpha,\beta)} v \otimes w,</math>
and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on ''v'' ⊗ ''W'' to a finite sum.

Specifically, if ''V'' is a highest weight module, then the formal infinite sum, ''R'', has a well-defined, and [[invertible]], action on ''V'' ⊗ ''V'', and this value of ''R'' (as an element of End(''V'' ⊗ ''V'')) satisfies the [[Yang–Baxter equation]], and therefore allows us to determine a representation of the [[braid group]], and to define quasi-invariants for [[knot (mathematics)|knots]], [[link (knot theory)|links]] and [[braid theory|braids]].

====Case 2: ''q'' is a root of unity====
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===Quantum groups at ''q'' = 0===
{{main|Crystal base}}

[[Masaki Kashiwara]] has researched the limiting behaviour of quantum groups as ''q'' → 0, and found a particularly well behaved base called a [[crystal base]].

===Description and classification by root-systems and Dynkin diagrams===
There has been considerable progress in describing finite quotients of quantum groups such as the above ''U<sub>q</sub>''('''g''') for ''q<sup>n</sup>'' = 1; one usually considers the class of '''pointed''' [[Hopf algebras]], meaning that all simple left or right comodules are 1-dimensional and thus the sum of all its simple subcoalgebras forms a group algebra called the '''coradical''':

* In 2002 H.-J. Schneider and N. Andruskiewitsch <ref>Andruskiewitsch, Schneider: Pointed Hopf algebras, New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.</ref> finished their classification of pointed Hopf algebras with an abelian co-radical group (excluding primes 2, 3, 5, 7), especially as the above finite quotients of ''U<sub>q</sub>''('''g''') decompose into ''E''′s (Borel part), dual ''F''′s and ''K''′s (Cartan algebra) just like ordinary [[Semisimple Lie algebra]]s:
::<math>\left(\mathfrak{B}(V)\otimes k[\mathbf{Z}^n]\otimes\mathfrak{B}(V^*)\right)^\sigma</math>
:Here, as in the classical theory ''V'' is a [[braided vector space]] of dimension ''n'' spanned by the ''E''′s, and ''σ'' (a so-called cocycle twist) creates the nontrivial '''linking''' between ''E''′s and ''F''′s. Note that in contrast to classical theory, more than two linked components may appear. The role of the '''quantum Borel algebra''' is taken by a [[Nichols algebra]] <math>\mathfrak{B}(V)</math> of the braided vectorspace. [[File:Dynkin4A3lift.png|thumb|generalized Dynkin diagram for a pointed Hopf algebra linking four A3 copies]]

* A crucial ingredient was I. Heckenberger's [[Nichols algebra|classification of finite Nichols algebras]] for abelian groups in terms of generalized [[Dynkin diagram]]s.<ref>Heckenberger: Nichols algebras of diagonal type and arithmetic root systems, Habilitation thesis 2005.</ref> When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dynkin diagram).
[[File:Dynkin Diagram Triangle.jpg|thumb|A rank 3 Dynkin diagram associated to a finite-dimensional Nichols algebra]]

* Meanwhile, Schneider and Heckenberger<ref>Heckenberger, Schneider: Root system and Weyl gruppoid for Nichols algebras, 2008.</ref> have generally proven the existence of an '''arithmetic''' [[root system]] also in the nonabelian case, generating a [[Poincaré–Birkhoff–Witt theorem|PBW basis]] as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used<ref>Heckenberger, Schneider: Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl grupoid, 2009.</ref> on specific cases  ''U<sub>q</sub>''('''g''') and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of the [[Weyl group]] of the [[Lie algebra]] '''g'''.