Group theory  
Group theory


In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra. There is no single, allencompassing definition, but instead a family of broadly similar objects.
The term "quantum group" often denotes a kind of noncommutative algebra with additional structure that first appeared in the theory of quantum integrable systems, and which was then formalized by Vladimir Drinfel'd and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a `bicrossproduct' class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo.
In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.
Just as groups often appear as symmetries, quantum groups act on many other mathematical objects and it has become fashionable to introduce the adjective quantum in such cases; for example there are quantum planes and quantum Grassmannians.
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The discovery of quantum groups was quite unexpected, since it was known for a long time that compact 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 a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of Alain Connes' noncommutative geometry. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum YangBaxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgenii Sklyanin, Nicolai Reshetikhin and 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 selfdual objects as an approach to quantum gravity^{[2]}.
One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfel'd and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a KacMoody algebra, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra.
Let A = (a_{ij}) be the Cartan matrix of the KacMoody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U_{q}(G), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators k_{λ} (where λ is an element of the weight lattice, i.e. for all i), and e_{i} and f_{i} (for simple roots, α_{i}), subject to the following relations:
where , , , for all positive integers n, and These are the qfactorial and qnumber, respectively, the qanalogs of the ordinary factorial. The last two relations above are the qSerre relations, the deformations of the Serre relations.
In the limit as , these relations approach the relations for the universal enveloping algebra U(G), where and as , where the element, t_{λ}, of the Cartan subalgebra satisfies (t_{λ},h) = λ(h) for all h in the Cartan subalgebra.
There are various coassociative coproducts under which these algebras are Hopf algebras, for example,
In addition, any Hopf algebra leads to another with reversed copproduct , where T is given by , giving three more possible versions.
The counit on U_{q}(A) is the same for all these coproducts: ε(k_{λ}) = 1, ε(e_{i}) = 0, ε(f_{i}) = 0, and the respective antipodes for the above coproducts are given by
Alternatively, the quantum group U_{q}(G)
can be regarded as an algebra over the field ,
the field of all rational functions of an
indeterminate q over .
Similarly, the quantum group U_{q}(G) can be regarded as an algebra over the field , the field of all rational functions of an indeterminate q over (see below in the section on quantum groups at q = 0).
Just as there are many different types of representations for KacMoody 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 an adjoint representation on itself as a module, with the action being given by
Ad_{x}.y =  ∑  x_{(1)}yS(x_{(2)}), 
(x) 
where .
One important type of representation is a weight representation, and the corresponding 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_{λ}.v = d_{λ}v for all λ, where d_{λ} are complex numbers for all weights λ such that
A weight module is called integrable if the actions of e_{i} and f_{i} are locally nilpotent (i.e. for any vector v in the module, there exists a positive integer k, possibly dependent on v, such that for all i). In the case of integrable modules, the complex numbers d_{λ} associated with a weight vector satisfy d_{λ} = c_{λ}q^{(λ,ν)}, where ν is an element of the weight lattice, and c_{λ} are complex numbers such that
Of special interest are highest weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vector v, subject to k_{λ}.v = d_{λ}v for all weights λ, and e_{i}.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_{λ}.v = d_{λ}v for all weights λ, and f_{i}.v = 0 for all i.
Define a vector v to have weight ν if k_{λ}.v = q^{(λ,ν)}v for all λ in the weight lattice.
If G is a KacMoody algebra, then in any irreducible highest weight representation of U_{q}(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 2(μ,α_{i}) / (α_{i},α_{i}) is a nonnegative 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 k_{λ}.v = c_{λ}q^{(λ,ν)}v, where c_{λ} are complex numbers such that
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_{q}(G), and for vectors v and w in the respective modules, , so that , and in the case of coproduct Δ_{1}, and .
The integrable highest weight module described above is a tensor product of a onedimensional module (on which k_{λ} = c_{λ} for all λ, and e_{i} = f_{i} = 0 for all i) and a highest weight module generated by a nonzero vector v_{0}, subject to k_{λ}.v_{0} = q^{(λ,ν)}v_{0} for all weights λ, and e_{i}.v_{0} = 0 for all i.
In the specific case where G is a finitedimensional Lie algebra (as a special case of a KacMoody algebra), then the irreducible representations with dominant integral highest weights are also finitedimensional.
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 KacMoody algebra (the highest weights are the same, as are their multiplicities).
Strictly, the quantum group U_{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 an Rmatrix. This infinite formal sum is expressible in terms of generators e_{i} and f_{i}, and Cartan generators t_{λ}, where k_{λ} is formally identified with . The infinite formal sum is the product of two factors, , and an infinite formal sum, where {λ_{j}} is a basis for the dual space to the Cartan subalgebra, and {μ_{j}} is the dual basis, and η is a sign (+1 or 1).
The formal infinite sum which plays the part of the Rmatrix has a welldefined action on the tensor product of two irreducible highest weight modules, and also on the tensor product if two lowest weight modules. Specifically, if v has weight α and w has weight β, then , and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on to a finite sum.
Specifically, if V is a highest weight module, then the formal infinite sum, R, has a welldefined, and invertible, action on , and this value of R (as an element of ) satisfies the YangBaxter equation, and therefore allows us to determine a representation of the braid group, and to define quasiinvariants for knots, links and braids.
Masaki Kashiwara has researched the limiting behaviour of quantum groups as .
As a consequence of the defining relations for the quantum group U_{q}(G), U_{q}(G) can be regarded as a Hopf algebra over , the field of all rational functions of an indeterminate q over .
For simple root α_{i} and nonnegative integer n, define and (specifically, ). In an integrable module M, and for weight λ, a vector (i.e. a vector u in M with weight λ) can be uniquely decomposed into the sums
where , , only if , and only if . Linear mappings and can be defined on M_{λ} by
Let A be the integral domain of all rational functions in which are regular at q = 0 (i.e. a rational function f(q) is an element of A if and only if there exist polynomials g(q) and h(q) in the polynomial ring such that , and f(q) = g(q) / h(q)). A crystal base for M is an ordered pair (L,B), such that
To put this into a more informal setting, the actions of e_{i}f_{i} and f_{i}e_{i} are generally singular at q = 0 on an integrable module M. The linear mappings and on the module are introduced so that the actions of and are regular at q = 0 on the module. There exists a basis of weight vectors for M, with respect to which the actions of and are regular at q = 0 for all i. The module is then restricted to the free Amodule generated by the basis, and the basis vectors, the Asubmodule and the actions of and are evaluated at q = 0. Furthermore, the basis can be chosen such that at q = 0, for all i, and are represented by mutual transposes, and map basis vectors to basis vectors or 0.
A crystal base can be represented by a directed graph with labelled edges. Each vertex of the graph represents an element of the basis B of L / qL, and a directed edge, labelled by i, and directed from vertex v_{1} to vertex v_{2}, represents that (and, equivalently, that ), where b_{1} is the basis element represented by v_{1}, and b_{2} is the basis element represented by v_{2}. The graph completely determines the actions of and at q = 0. If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets V_{1} and V_{2} such that there are no edges joining any vertex in V_{1} to any vertex in V_{2}).
For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate KacMoody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate KacMoody algebra.
It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base.
Let M be an integrable module with crystal base (L,B) and M' be an integrable module with crystal base (L',B'). For crystal bases, the coproduct Δ, given by , is adopted. The integrable module has crystal base , where . For a basis vector , define and . The actions of and on are given by
The decomposition of the product two integrable highest weight modules into irreducible submodules is determined by the decomposition of the graph of the crystal base into its connected components (i.e. the highest weights of the submodules are determined, and the multiplicity of each highest weight is determined).
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 a C*algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.
The continuous complexvalued functions on a compact Hausdorff topological space form a commutative C*algebra. By the Gelfand theorem, a commutative C*algebra is isomorphic to the C*algebra of continuous complexvalued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*algebra up to homeomorphism.
For a compact topological group, G, there exists a C*algebra homomorphism (where is the C*algebra tensor product  the completion of the algebraic tensor product of C(G) and C(G)), such that Δ(f)(x,y) = f(xy) for all , and for all (where for all and all ). There also exists a linear multiplicative mapping , such that κ(f)(x) = f(x ^{− 1}) for all and all . Strictly, this does not make C(G) a Hopf algebra, unless G is finite. On the other hand, a finitedimensional representation of G can be used to generate a *subalgebra of C(G) which is also a Hopf *algebra. Specifically, if is an ndimensional representation of G, then for all i,j, and for all i,j. It follows that the *algebra generated by u_{ij} for all i,j and κ(u_{ij}) for all i,j is a Hopf *algebra: the counit is determined by ε(u_{ij}) = δ_{ij} for all i,j (where δ_{ij} is the Kronecker delta), the antipode is κ, and the unit is given by
1 =  ∑  u_{1k}κ(u_{k1}) =  ∑  κ(u_{1k})u_{k1}. 
k  k 
As a generalization, a compact matrix quantum group is defined as a pair (C,u), where C is a C*algebra and is a matrix with entries in C such that
∑  κ(u_{ik})u_{k j} =  ∑  u_{ik}κ(u_{k j}) = δ_{ij}I, 
k  k 
As a consequence of continuity, the comultiplication on C is coassociative.
In general, C is not a bialgebra, and C_{0} is a Hopf *algebra.
Informally, C can be regarded as the *algebra of continuous complexvalued functions over the compact matrix quantum group, and u can be regarded as a finitedimensional representation of the compact matrix quantum group.
A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *algebra (a corepresentation of a counital coassiative coalgebra A is a square matrix with entries in A (so ) such that for all i,j and ε(v_{ij}) = δ_{ij} for all i,j). Furthermore, a representation, v, is called unitary if the matrix for v is unitary (or equivalently, if for all i, 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
and so that the comultiplication is determined by , , and the coinverse is determined by κ(α) = α ^{*} , κ(γ) = − μ ^{− 1}γ, κ(γ ^{*} ) = − μγ ^{*} , κ(α ^{*} ) = α. Note that u 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
and so that the comultiplication is determined by , , and the coinverse is determined by κ(α) = α ^{*} , κ(β) = − μ ^{− 1}β, κ(β ^{*} ) = − μβ ^{*} , κ(α ^{*} ) = α. Note that w is a unitary representation. The realizations can be identified by equating .
When μ = 1, then SU_{μ}(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).
Whereas compact matrix pseudogroups are typically versions of DrinfeldJimbo 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 of locally acting on each other and results in a quantum group (given here in an algebraic form) with generators p,K,K ^{− 1}, say, and coproduct
[p,K] = hK(K − 1), ,
where h is the deformation parameter. This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra (the Iwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated to g. For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions.
