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. Distinct but related objects, also called quantum groups, are deformations of the algebra of functions on a semisimple algebraic group or a compact Lie group.
Since the discovery of quantum groups, it has become fashionable to introduce the attribute quantum into the names of many other mathematical objects, such as quantum plane or quantum grassmanian. They may also be loosely referred to as aspects of "quantum groups".
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, cannot be deformed. One of the ideas behind quantum groups is that if we consider in some sense equivalent but larger structure, namely a group algebra or a universal enveloping algebra, then it 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 Yang-Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgenii Sklyanin, Nicolai Reshetikhin and others) and related work by the Japanese School.
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 Kac-Moody algebra, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra.
Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, , where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators (where is an element of the weight lattice, i.e. for all i), and and (for simple roots, ), subject to the following relations:
where , , , for all positive integers , and . These are the q-factorial and q-number, respectively, the q-analogs of the ordinary factorial. The last two relations above are the q-Serre relations, the deformations of the Serre relations.
In the limit as , these relations approach the relations for the universal enveloping algebra , where and as , where the element, , of the Cartan subalgebra satisfies for all h in the Cartan subalgebra.
There are various coassociative coproducts under which the quantum groups are Hopf algebras, for example,
along with the reverse coproducts , where is given by , i.e.
Alternatively, the quantum group can be regarded as an algebra over the field , the field of all rational functions of an indeterminate q over .
Similarly, the quantum group 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 representation 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, has an adjoint representation on itself as a module, with the action being given by , 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 for all , where are complex numbers for all weights such that
A weight module is called integrable if the actions of and 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 associated with a weight vector satisfy , where is an element of the weight lattice, and 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 for all weights , and 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 for all weights , and for all i.
Define a vector v to have weight if for all in the weight lattice.
If G is a Kac-Moody algebra, then in any irreducible highest weight representation of , with highest weight , the multiplicities of the weights are equal to their multiplicities in an irreducible representation of with equal highest weight. If the highest weight is dominant and integral (a weight is dominant and integral if satisfies the condition that 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 , where are complex numbers such that
and 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 , and for vectors v and w in the respective modules, , so that , and in the case of coproduct , and .
The integrable highest weight module described above is a tensor product of a one-dimensional module (on which for all , and for all i) and a highest weight module generated by a nonzero vector , subject to for all weights , and 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).
Strictly, the quantum group 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. This infinite formal sum is expressible in terms of generators and , and Cartan generators , where is formally identified with . The infinite formal sum is the product of two factors, , and an infinite formal sum, where is a basis for the dual space to the Cartan subalgebra, and is the dual basis, and is a sign (+1 or -1).
The formal infinite sum which plays the part of the R-matrix has a well-defined 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 and w has weight , 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 well-defined, and invertible, action on , and this value of R (as an element of ) satisfies the Yang-Baxter equation, and therefore allows us to determine a representation of the braid group, and to define quasi-invariants 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 , can be regarded as a Hopf algebra over , the field of all rational functions of an indeterminate q over .
For simple root and non-negative integer , define and (specifically, ). In an integrable module , and for weight , a vector (i.e. a vector in with weight ) can be uniquely decomposed into the sums
where , , only if , and only if . Linear mappings and can be defined on by
Let be the integral domain of all rational functions in which are regular at (i.e. a rational function is an element of if and only if there exist polynomials and in the polynomial ring such that , and ). A crystal base for is an ordered pair , such that
To put this into a more informal setting, the actions of and are generally singular at on an integrable module . The linear mappings and on the module are introduced so that the actions of and are regular at on the module. There exists a -basis of weight vectors for , with respect to which the actions of and are regular at for all i. The module is then restricted to the free -module generated by the basis, and the basis vectors, the -submodule and the actions of and are evaluated at . Furthermore, the basis can be chosen such that at , for all , 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 of , and a directed edge, labelled by i, and directed from vertex to vertex , represents that (and, equivalently, that ), where is the basis element represented by , and is the basis element represented by . The graph completely determines the actions of and at . 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 and such that there are no edges joining any vertex in to any vertex in ).
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 Kac-Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac-Moody 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 be an integrable module with crystal base and be an integrable module with crystal base . For crystal bases, the coproduct , 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 complex-valued 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 complex-valued 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 and ), such that for all , and for all (where for all and all ). There also exists a linear multiplicative mapping , such that for all and all . Strictly, this does not make a Hopf algebra, unless G is finite. On the other hand, a finite-dimensional representation of G can be used to generate a *-subalgebra of which is also a Hopf *-algebra. Specifically, if is an -dimensional representation of , then for all , and for all . It follows that the *-algebra generated by for all and for all is a Hopf *-algebra: the counit is determined by for all (where is the Kronecker delta), the antipode is , and the unit is given by
As a generalization, a compact matrix quantum group is defined as a pair , where is a C*-algebra and is a matrix with entries in such that
As a consequence of continuity, the comultiplication on is coassociative.
In general, is not a bialgebra, and is a Hopf *-algebra.
Informally, can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and 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 a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassiative coalgebra is a square matrix with entries in (so ) such that for all and for all ). 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 , where the parameter is a positive real number. So , where is the C*-algebra generated by and ,subject to
and so that the comultiplication is determined by , , and the coinverse is determined by , , , . Note that is a representation, but not a unitary representation. is equivalent to the unitary representation
Equivalently, , where is the C*-algebra generated by and ,subject to
and so that the comultiplication is determined by , , and the coinverse is determined by , , , . Note that is a unitary representation. The realizations can be identified by equating .
When , then is equal to the concrete compact group .