Quantum groups

The phrase "quantum group" is more or less a synonym for "Hopf algebra" . More precisely, the category of quantum groups is defined in [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 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 with an associative operation one obtains a commutative algebra over a commutative ring with a homomorphism , called comultiplication; the unit gives rise to a homomorphism , called co-unit, and the mapping , , gives rise to a bijective -linear mapping , called antipode. The group axioms are equivalent to the commutativity of the following diagrams:

Here , . The commutativity of these diagrams means that 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. is contained in the symmetric part of . Essentially, all cocommutative Hopf algebras are group algebras.

Here is an example of a Hopf algebra which is neither commutative nor cocommutative. Fix and , where is a commutative ring. Denote by the associative -algebra with generators , , and defining relations if , if , if , , if , , , where is the number of inversions in the permutation . Then has a Hopf algebra structure defined by . If , then is the algebra of polynomial functions on . So, in the general case it is natural to consider elements of as "functions on the quantized SLn" .

The quantized 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 [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 of a commutative Hopf algebra is given, then a Poisson bracket on is defined by , where is the deformation parameter and means the deformed product, which is not commutative. This Poisson bracket has the usual properties (skew-symmetry, Jacobi identity, ) and is compatible with comultiplication. In other words, is a Poisson–Hopf algebra. Therefore it is natural to start with a Poisson–Hopf algebra and then try to quantize it, i.e. to construct a Hopf algebra deformation of which induces the given Poisson bracket on .

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 [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 with a linear mapping such that: 1) defines a Lie algebra structure on ; and 2) is a -cocycle ( acts on by means of the adjoint representation). By definition, a quantization of is a Hopf algebra deformation of the universal enveloping algebra such that , where is the Poisson cobracket, defined by . Here is the deformation parameter, the deformed comultiplication and 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. [a1], §3, §6) there exists a canonical quantization. In particular, on a Kac–Moody algebra with a fixed scalar product there is a canonical Lie bi-algebra structure and this bi-algebra has a canonical quantization , as was discovered in [a3], [a4], [a5]. Let be the Cartan subalgebra of , the images of the simple roots . Then is generated by and , with the following defining relations:

Setting , one has also

Here is the Cartan matrix and is the Gauss polynomial, i.e.,

The comultiplication in is such that for and

If is a finite-dimensional simple Lie algebra (cf. Lie algebra, semi-simple), then the algebra of regular functions on the corresponding simply-connected algebraic group is isomorphic to the subalgebra of generated by the matrix elements of the finite-dimensional representations of . Therefore the subalgebra of generated by the matrix elements of the finite-dimensional representations of can be considered as the algebra of functions on a certain quantization of . For instance, the quantized (cf. above) can be obtained in this way.

There is an important notion of a quasitriangular Hopf algebra. This is a pair where is a Hopf algebra and is an invertible element of such that , , for . Here is the opposite comultiplication and , , are defined as follows: If , where , then , , . If is a quasitriangular Hopf algebra, then satisfies the quantum Yang–Baxter equation (cf. also Yang–Baxter equation), i.e., . It is known (cf. [a1], §13) that if is a finite-dimensional simple Lie algebra, then has a canonical quasitriangular structure, while if is an infinite-dimensional Kac–Moody algebra, then has an "almost quasitriangular" structure.

If is a quasitriangular Hopf algebra over and is a representation , then satisfies the quantum Yang–Baxter equation. There is an inverse construction (cf. [a6], [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 , , with defining relations , where , , is the matrix , and is defined by .

Quasitriangular Hopf algebras are a natural tool for the quantum inverse-scattering in method ([a1], §11). On the other hand, they can be used (cf. [a8]) to construct invariants of knots (and of more general objects such as links and tangles) generalizing the Jones polynomial [a9]. More precisely, to an oriented knot and a quasitriangular Hopf algebra there corresponds a central element .

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 [a10] (the idea is to use -algebras instead of abstract algebras). The quantized (cf. [a11], [a12]) is a typical example. The notion of a ring group (cf. , [a14]) and the equivalent notion of a Kac algebra (cf. [a15], [a16]) were introduced as an attempt to define a locally compact quantum group. However, these notions are not general enough (the axioms of , [a14], [a15] imply that the square of the antipode is the identity mapping, and therefore the quantized is not a ring group).

References

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