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Inmathematics andtheoretical physics, the termquantum group denotes one of a few different kinds ofnoncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which arequasitriangular Hopf algebras),compact matrix quantum groups (which are structures on unital separableC*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group.

The term "quantum group" first appeared in the theory ofquantum integrable systems, which was then formalized byVladimir Drinfeld andMichio Jimbo as a particular class ofHopf algebra. The same term is also used for other Hopf algebras that deform or are close to classicalLie groups orLie algebras, such as a "bicrossproduct" class of quantum groups introduced byShahn Majid a little after the work of Drinfeld and Jimbo.

In Drinfeld's approach, quantum groups arise asHopf algebras depending on an auxiliary parameterq orh, which becomeuniversal enveloping algebras of a certain Lie algebra, frequentlysemisimple oraffine, whenq = 1 orh = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimplealgebraic group or acompact Lie group.

The discovery of quantum groups was quite unexpected since it was known for a long time thatcompact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely agroup algebra or auniversal enveloping algebra, then a group algebra or enveloping algebra can be "deformed", although the deformation will no longer remain a group algebra or enveloping algebra. More precisely, deformation can be accomplished within the category ofHopf algebras that are not required to be eithercommutative orcocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of thenoncommutative geometry ofAlain Connes. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantumYang–Baxter equation andquantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev,Leon Takhtajan,Evgeny Sklyanin,Nicolai Reshetikhin andVladimir Korepin) and related work by the Japanese School.^[1] The intuition behind the second,bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach toquantum gravity.^[2]

One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of theuniversal enveloping algebra of asemisimple Lie algebra or, more generally, aKac–Moody algebra, in the category ofHopf algebras. The resulting algebra has additional structure, making it into aquasitriangular Hopf algebra.

LetA = (a_ij) be theCartan matrix of the Kac–Moody algebra, and letq ≠ 0, 1 be a complex number, then the quantum group,U_q(G), whereG is the Lie algebra whose Cartan matrix isA, is defined as theunitalassociative algebra with generatorsk_λ (whereλ is an element of theweight lattice, i.e. 2(λ, α_i)/(α_i, α_i) is an integer for alli), ande_i andf_i (forsimple roots, α_i), subject to the following relations:

And fori ≠j we have theq-Serre relations, which are deformations of theSerre relations:

where theq-factorial, theq-analog of the ordinaryfactorial, is defined recursively using q-number:

In the limit asq → 1, these relations approach the relations for the universal enveloping algebraU(G), where

${\displaystyle k_{\lambda }\to 1,\frac{k_{\lambda }-k_{-\lambda }}{q-q^{-1}}\to t_{\lambda }}$

andt_λ is the element of the Cartan subalgebra satisfying (t_λ,h) =λ(h) for allh in the Cartan subalgebra.

There are variouscoassociative coproducts under which these algebras are Hopf algebras, for example,

where the set of generators has been extended, if required, to includek_λ forλ which is expressible as the sum of an element of the weight lattice and half an element of theroot lattice.

In addition, any Hopf algebra leads to another with reversed coproductT o Δ, whereT is given byT(x ⊗y) =y ⊗x, giving three more possible versions.

Thecounit onU_q(A) is the same for all these coproducts:ε(k_λ) = 1,ε(e_i) =ε(f_i) = 0, and the respectiveantipodes for the above coproducts are given by

Alternatively, the quantum groupU_q(G) can be regarded as an algebra over the fieldC(q), the field of allrational functions of an indeterminateq overC.

Similarly, the quantum groupU_q(G) can be regarded as an algebra over the fieldQ(q), the field of allrational functions of an indeterminateq overQ (see below in the section on quantum groups atq = 0). The center of quantum group can be described by quantum determinant.

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_q(G) has anadjoint representation on itself as a module, with the action being given by

${\displaystyle {\mathrm{A}\mathrm{d}}_x\cdot y=\sum _{(x)}{x}_{(1)}yS(x_{(2)}),}$

where

${\displaystyle \mathrm{\Delta }(x)=\sum _{(x)}{x}_{(1)}\otimes x_{(2)}.}$

One important type of representation is a weight representation, and the correspondingmodule is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vectorv such thatk_λ ·v =d_λv for allλ, whered_λ are complex numbers for all weightsλ such that

${\displaystyle d_0=1,}$

${\displaystyle d_{\lambda }{d}_{\mu }=d_{\lambda +\mu },}$ for all weightsλ andμ.

A weight module is called integrable if the actions ofe_i andf_i are locally nilpotent (i.e. for any vectorv in the module, there exists a positive integerk, possibly dependent onv, such that${\displaystyle e_i^k.v=f_i^k.v=0}$ for alli). In the case of integrable modules, the complex numbersd_λ associated with a weight vector satisfy${\displaystyle d_{\lambda }=c_{\lambda }{q}^{(\lambda ,\nu )}}$,^{[citation needed]} whereν is an element of the weight lattice, andc_λ are complex numbers such that

• ${\displaystyle c_0=1,}$
• ${\displaystyle c_{\lambda }{c}_{\mu }=c_{\lambda +\mu },}$ for all weightsλ andμ,
• ${\displaystyle c_{2{\alpha }_i}=1}$ for alli.

Of special interest arehighest-weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vectorv, subject tok_λ ·v =d_λv for all weightsμ, ande_i ·v = 0 for alli. Similarly, a quantum group can have a lowest weight representation and lowest weight module,i.e. a module generated by a weight vectorv, subject tok_λ ·v =d_λv for all weightsλ, andf_i ·v = 0 for alli.

Define a vectorv to have weightν if${\displaystyle k_{\lambda }\cdot v=q^{(\lambda ,\nu )}v}$ for allλ in the weight lattice.

IfG is a Kac–Moody algebra, then in any irreducible highest weight representation ofU_q(G), with highest weight ν, the multiplicities of the weights are equal to their multiplicities in an irreducible representation ofU(G) with equal highest weight. If the highest weight is dominant and integral (a weightμ is dominant and integral ifμ satisfies the condition that${\displaystyle 2(\mu ,{\alpha }_i)/({\alpha }_i,{\alpha }_i)}$ is a non-negative integer for alli), then the weight spectrum of the irreducible representation is invariant under theWeyl group forG, and the representation is integrable.

Conversely, if a highest weight module is integrable, then its highest weight vectorv satisfies${\displaystyle k_{\lambda }\cdot v=c_{\lambda }{q}^{(\lambda ,\nu )}v}$, wherec_λ ·v =d_λv are complex numbers such that

• ${\displaystyle c_0=1,}$
• ${\displaystyle c_{\lambda }{c}_{\mu }=c_{\lambda +\mu },}$ for all weightsλ andμ,
• ${\displaystyle c_{2{\alpha }_i}=1}$ for alli,

andν is dominant and integral.

As is the case for all Hopf algebras, thetensor product of two modules is another module. For an elementx ofU_q(G), and for vectorsv andw in the respective modules,x ⋅ (v ⊗w) = Δ(x) ⋅ (v ⊗w), so that${\displaystyle k_{\lambda }\cdot (v\otimes w)=k_{\lambda }\cdot v\otimes k_{\lambda }.w}$, and in the case of coproduct Δ₁,${\displaystyle e_i\cdot (v\otimes w)=k_i\cdot v\otimes e_i\cdot w+e_i\cdot v\otimes w}$ and${\displaystyle f_i\cdot (v\otimes w)=v\otimes f_i\cdot w+f_i\cdot v\otimes k_i^{-1}\cdot w.}$

The integrable highest weight module described above is a tensor product of a one-dimensional module (on whichk_λ =c_λ for allλ, ande_i =f_i = 0 for alli) and a highest weight module generated by a nonzero vectorv₀, subject to${\displaystyle k_{\lambda }\cdot v_0=q^{(\lambda ,\nu )}{v}_0}$ for all weightsλ, and${\displaystyle e_i\cdot v_0=0}$ for alli.

In the specific case whereG 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).

Strictly, the quantum groupU_q(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 anR-matrix. This infinite formal sum is expressible in terms of generatorse_i andf_i, and Cartan generatorst_λ, wherek_λ is formally identified withq^t_λ. The infinite formal sum is the product of two factors,^{[citation needed]}

${\displaystyle q^{\eta \sum _j{t}_{{\lambda }_j}\otimes t_{{\mu }_j}}}$

and an infinite formal sum, whereλ_j is a basis for the dual space to the Cartan subalgebra, andμ_j is the dual basis, andη = ±1.

The formal infinite sum which plays the part of theR-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, ifv has weightα andw has weightβ, then

${\displaystyle q^{\eta \sum _j{t}_{{\lambda }_j}\otimes t_{{\mu }_j}}\cdot (v\otimes w)=q^{\eta (\alpha ,\beta )}v\otimes w,}$

and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor onv ⊗W to a finite sum.

Specifically, ifV is a highest weight module, then the formal infinite sum,R, has a well-defined, andinvertible, action onV ⊗V, and this value ofR (as an element of End(V ⊗V)) satisfies theYang–Baxter equation, and therefore allows us to determine a representation of thebraid group, and to define quasi-invariants forknots,links andbraids.

Masaki Kashiwara has researched the limiting behaviour of quantum groups asq → 0, and found a particularly well behaved base called acrystal base.

There has been considerable progress in describing finite quotients of quantum groups such as the aboveU_q(g) forqⁿ = 1; one usually considers the class ofpointedHopf 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 thecoradical:

• In 2002 H.-J. Schneider and N. Andruskiewitsch^[3] 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 ofU_q(g) decompose intoE′s (Borel part), dualF′s andK′s (Cartan algebra) just like ordinarySemisimple Lie algebras:

${\displaystyle {\left(\mathfrak{B}(V)\otimes k[{\mathbf{Z}}^n]\otimes \mathfrak{B}(V^{\ast })\right)}^{\sigma }}$

Here, as in the classical theoryV is abraided vector space of dimensionn spanned by theE′s, andσ (a so-called cocycle twist) creates the nontriviallinking betweenE′s andF′s. Note that in contrast to classical theory, more than two linked components may appear. The role of thequantum Borel algebra is taken by aNichols algebra${\displaystyle \mathfrak{B}(V)}$ of the braided vectorspace.

• A crucial ingredient was I. Heckenberger'sclassification of finite Nichols algebras for abelian groups in terms of generalizedDynkin diagrams.^[4] When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dynkin diagram).

• Meanwhile, Schneider and Heckenberger^[5] have generally proven the existence of anarithmeticroot system also in the nonabelian case, generating aPBW basis as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used^[6] on specific casesU_q(g) and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of theWeyl group of theLie algebrag.

S. L. Woronowicz introduced compact matrix quantum groups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of aC*-algebra. The geometry of a compact matrix quantum group is a special case of anoncommutative geometry.

The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. By theGelfand theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up tohomeomorphism.

For a compacttopological group,G, there exists a C*-algebra homomorphism Δ:C(G) →C(G) ⊗C(G) (whereC(G) ⊗C(G) is the C*-algebra tensor product - the completion of the algebraic tensor product ofC(G) andC(G)), such that Δ(f)(x,y) =f(xy) for allf ∈C(G), and for allx,y ∈G (where (f ⊗g)(x,y) =f(x)g(y) for allf,g ∈C(G) and allx,y ∈G). There also exists a linear multiplicative mappingκ:C(G) →C(G), such thatκ(f)(x) =f(x⁻¹) for allf ∈C(G) and allx ∈G. Strictly, this does not makeC(G) a Hopf algebra, unlessG is finite. On the other hand, a finite-dimensionalrepresentation ofG can be used to generate a *-subalgebra ofC(G) which is also a Hopf *-algebra. Specifically, if${\displaystyle g\mapsto (u_{ij}(g){)}_{i,j}}$ is ann-dimensional representation ofG, then for alli,ju_ij ∈C(G) and

${\displaystyle \mathrm{\Delta }(u_{ij})=\sum _k{u}_{ik}\otimes u_{kj}.}$

It follows that the *-algebra generated byu_ij for alli, j andκ(u_ij) for alli, j is a Hopf *-algebra: the counit is determined by ε(u_ij) = δ_ij for alli, j (whereδ_ij is theKronecker delta), the antipode isκ, and the unit is given by

${\displaystyle 1=\sum _k{u}_{1k}\kappa (u_{k1})=\sum _k\kappa (u_{1k})u_{k1}.}$

As a generalization, a compact matrix quantum group is defined as a pair (C,u), whereC is a C*-algebra and${\displaystyle u=(u_{ij}{)}_{i,j=1,\dots ,n}}$ is a matrix with entries inC such that

• The *-subalgebra,C₀, ofC, which is generated by the matrix elements ofu, is dense inC;

• There exists a C*-algebra homomorphism called the comultiplication Δ:C →C ⊗C (whereC ⊗C is the C*-algebra tensor product - the completion of the algebraic tensor product ofC andC) such that for alli, j we have:

${\displaystyle \mathrm{\Delta }(u_{ij})=\sum _k{u}_{ik}\otimes u_{kj}}$

• There exists a linear antimultiplicative map κ:C₀ →C₀ (the coinverse) such thatκ(κ(v*)*) =v for allv ∈C₀ and

${\displaystyle \sum _k\kappa (u_{ik})u_{kj}=\sum _k{u}_{ik}\kappa (u_{kj})={\delta }_{ij}I,}$

whereI is the identity element ofC. Since κ is antimultiplicative, thenκ(vw) =κ(w)κ(v) for allv,w inC₀.

As a consequence of continuity, the comultiplication onC is coassociative.

In general,C is not a bialgebra, andC₀ is a Hopf *-algebra.

Informally,C can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, andu can be regarded as a finite-dimensional representation of the compact matrix quantum group.

A representation of the compact matrix quantum group is given by acorepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebraA is a square matrix${\displaystyle v=(v_{ij}{)}_{i,j=1,\dots ,n}}$ with entries inA (sov belongs to M(n,A)) such that

${\displaystyle \mathrm{\Delta }(v_{ij})=\sum _{k=1}^n{v}_{ik}\otimes v_{kj}}$

for alli,j andε(v_ij) = δ_ij for alli, j). Furthermore, a representationv, is called unitary if the matrix forv is unitary (or equivalently, if κ(v_ij) =v*_ij for alli,j).

An example of a compact matrix quantum group is SU_μ(2), where the parameter μ is a positive real number. So SU_μ(2) = (C(SU_μ(2)),u), where C(SU_μ(2)) is the C*-algebra generated by α and γ, subject to

${\displaystyle \gamma {\gamma }^{\ast }={\gamma }^{\ast }\gamma ,}$

${\displaystyle \alpha \gamma =\mu \gamma \alpha ,}$

${\displaystyle \alpha {\gamma }^{\ast }=\mu {\gamma }^{\ast }\alpha ,}$

${\displaystyle \alpha {\alpha }^{\ast }+\mu {\gamma }^{\ast }\gamma ={\alpha }^{\ast }\alpha +{\mu }^{-1}{\gamma }^{\ast }\gamma =I,}$

and

so that the comultiplication is determined by ∆(α) = α ⊗ α − γ ⊗ γ*, ∆(γ) = α ⊗ γ + γ ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(γ) = −μ⁻¹γ, κ(γ*) = −μγ*, κ(α*) = α. Note thatu is a representation, but not a unitary representation.u is equivalent to the unitary representation

Equivalently, SU_μ(2) = (C(SU_μ(2)),w), where C(SU_μ(2)) is the C*-algebra generated by α and β, subject to

${\displaystyle \beta {\beta }^{\ast }={\beta }^{\ast }\beta ,}$

${\displaystyle \alpha \beta =\mu \beta \alpha ,}$

${\displaystyle \alpha {\beta }^{\ast }=\mu {\beta }^{\ast }\alpha ,}$

${\displaystyle \alpha {\alpha }^{\ast }+{\mu }^2{\beta }^{\ast }\beta ={\alpha }^{\ast }\alpha +{\beta }^{\ast }\beta =I,}$

and

so that the comultiplication is determined by ∆(α) = α ⊗ α − μβ ⊗ β*, Δ(β) = α ⊗ β + β ⊗ α*, and the coinverse is determined by κ(α) = α*, κ(β) = −μ⁻¹β, κ(β*) = −μβ*, κ(α*) = α. Note thatw is a unitary representation. The realizations can be identified by equating${\displaystyle \gamma =\sqrt{\mu }\beta }$.

When μ = 1, then SU_μ(2) is equal to the algebraC(SU(2)) of functions on the concrete compact group SU(2).

Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first.

The very simplest nontrivial example corresponds to two copies ofR locally acting on each other and results in a quantum group (given here in an algebraic form) with generatorsp,K,K⁻¹, say, and coproduct

${\displaystyle [p,K]=hK(K-1)}$

${\displaystyle \mathrm{\Delta }p=p\otimes K+1\otimes p}$

${\displaystyle \mathrm{\Delta }K=K\otimes K}$

whereh is the deformation parameter.

This quantum group was linked to a toy model of Planck scale physics implementingBorn reciprocity when viewed as a deformation of theHeisenberg algebra of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebrag its complexification as a real Lie algebra of twice the dimension splits intog and a certain solvable Lie algebra (theIwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated tog. Forsu(2) one obtains a quantum group deformation of theEuclidean group E(3) of motions in 3 dimensions.

• Hopf algebra
• Lie bialgebra
• Poisson–Lie group
• Quantum affine algebra

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• Quantum groups
• Mathematical quantization

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