Quantum group

In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure.

The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld 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.

More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative.

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, Evgeny Sklyanin, Nicolai Reshetikhin and Vladimir Korepin) and related work by the Japanese School.

Let A = (aij) be the Cartan matrix of the Kac–Moody algebra, and let q ≠ 0, 1 be a complex number, then the quantum group, Uq(G), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators kλ (where λ is an element of the weight lattice, i.e. 2(λ, αi)/(αi, αi) is an integer for all i), and ei and fi (for simple roots, αi), subject to the following relations: And for i ≠ j we have the q-Serre relations, which are deformations of the Serre relations: where the q-factorial, the q-analog of the ordinary factorial, is defined recursively using q-number: In the limit as q → 1, these relations approach the relations for the universal enveloping algebra U(G), where and tλ is the element of the Cartan subalgebra satisfying (tλ, h) = λ(h) for all h in the Cartan subalgebra.

There are various coassociative coproducts under which these algebras are Hopf algebras, for example, where the set of generators has been extended, if required, to include kλ 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 o Δ, where T is given by T(x ⊗ y) = y ⊗ x, giving three more possible versions.

The counit on Uq(A) is the same for all these coproducts: ε(kλ) = 1, ε(ei) = ε(fi) = 0, and the respective antipodes for the above coproducts are given by Alternatively, the quantum group Uq(G) can be regarded as an algebra over the field C(q), the field of all rational functions of an indeterminate q over C. Similarly, the quantum group Uq(G) can be regarded as an algebra over the field Q(q), the field of all rational functions of an indeterminate q over Q (see below in the section on quantum groups at q = 0).

A weight vector is a nonzero vector v such that kλ · v = dλv for all λ, where dλ are complex numbers for all weights λ such that A weight module is called integrable if the actions of ei and fi are locally nilpotent (i.e. for any vector v in the module, there exists a positive integer k, possibly dependent on v, such that

In the case of integrable modules, the complex numbers dλ associated with a weight vector satisfy

,[citation needed] where ν is an element of the weight lattice, and cλ are complex numbers such that Of special interest are highest-weight representations, and the corresponding highest weight modules.

, where cλ · v = dλv are complex numbers such that and ν is dominant and integral.

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).

This infinite formal sum is expressible in terms of generators ei and fi, and Cartan generators tλ, where kλ is formally identified with qtλ.

The infinite formal sum is the product of two factors,[citation needed] and an infinite formal sum, where λj is a basis for the dual space to the Cartan subalgebra, and μj is the dual basis, and η = ±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 of two lowest weight modules.

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 knots, links and braids.

There has been considerable progress in describing finite quotients of quantum groups such as the above Uq(g) for qn = 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: S. L. Woronowicz introduced compact matrix quantum groups.

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 Δ: C(G) → C(G) ⊗ C(G) (where C(G) ⊗ C(G) 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 f ∈ C(G), and for all x, y ∈ G (where (f ⊗ g)(x, y) = f(x)g(y) for all f, g ∈ C(G) and all x, y ∈ G).

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

An example of a compact matrix quantum group is SUμ(2), where the parameter μ is a positive real number.

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 of R 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 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.