Quantum group
Algebraic structure → Group theory Group theory |
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In
The term "quantum group" first appeared in the theory of
In Drinfeld's approach, quantum groups arise as
Intuitive meaning
The discovery of quantum groups was quite unexpected since it was known for a long time that
Drinfeld–Jimbo type quantum groups
One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld 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 A = (aij) be the
And for i ≠ j we have the q-Serre relations, which are deformations of the Serre relations:
where the
, 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
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
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). The center of quantum group can be described by quantum determinant.
Representation theory
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, Uq(G) has an
where
Case 1: q is not a root of unity
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 kλ · v = dλv for all λ, where dλ are complex numbers for all weights λ such that
- for all weights λ and μ.
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 for all i). 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
- for all weights λ and μ,
- for all i.
Of special interest are
Define a vector v to have weight ν if for all λ in the weight lattice.
If G is a Kac–Moody algebra, then in any irreducible highest weight representation of Uq(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 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 cλ · v = dλv are complex numbers such that
- for all weights λ and μ,
- for all i,
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 Uq(G), and for vectors v and w in the respective modules, x ⋅ (v ⊗ w) = Δ(x) ⋅ (v ⊗ w), so that , and in the case of coproduct Δ1, and
The integrable highest weight module described above is a tensor product of a one-dimensional module (on which kλ = cλ for all λ, and ei = fi = 0 for all i) and a highest weight module generated by a nonzero vector v0, subject to for all weights λ, 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).
Case 2: q is a root of unity
Quasitriangularity
Case 1: q is not a root of unity
Strictly, the quantum group Uq(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 R-matrix. 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 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 v ⊗ W to a finite sum.
Specifically, if V is a highest weight module, then the formal infinite sum, R, has a well-defined, and
Case 2: q is a root of unity
Quantum groups at q = 0
Masaki Kashiwara has researched the limiting behaviour of quantum groups as q → 0, and found a particularly well behaved base called a crystal base.
Description and classification by root-systems and Dynkin diagrams
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
- 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 of Uq(g) decompose into E′s (Borel part), dual F′s and K′s (Cartan algebra) just like ordinary Semisimple Lie algebras:
- Here, as in the classical theory V is a braided vector space of dimension n spanned by the E′s, and σ (a so-called cocycle twist) creates the nontrivial linking between E′s and F′s. Note that in contrast to classical theory, more than two linked components may appear. The role of the quantum Borel algebra is taken by a Nichols algebra of the braided vectorspace.
- A crucial ingredient was I. Heckenberger's classification of finite Nichols algebras for abelian groups in terms of generalized Dynkin diagrams.[4] When small primes are present, some exotic examples, such as a triangle, occur (see also the Figure of a rank 3 Dankin diagram).
- Meanwhile, Schneider and Heckenberger[5] have generally proven the existence of an arithmetic root system also in the nonabelian case, generating a PBW basis as proven by Kharcheko in the abelian case (without the assumption on finite dimension). This can be used[6] on specific cases Uq(g) and explains e.g. the numerical coincidence between certain coideal subalgebras of these quantum groups and the order of the Weyl group of the Lie algebra g.
Compact matrix quantum groups
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 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). There also exists a linear multiplicative mapping κ: C(G) → C(G), such that κ(f)(x) = f(x−1) for all f ∈ C(G) and all x ∈ G. Strictly, this does not make C(G) 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 C(G) which is also a Hopf *-algebra. Specifically, if is an n-dimensional representation of G, then for all i, j uij ∈ C(G) and
It follows that the *-algebra generated by uij for all i, j and κ(uij) for all i, j is a Hopf *-algebra: the counit is determined by ε(uij) = δij for all i, j (where δij is the Kronecker delta), the antipode is κ, and the unit is given by
General definition
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
- The *-subalgebra, C0, of C, which is generated by the matrix elements of u, is dense in C;
- There exists a C*-algebra homomorphism called the comultiplication Δ: C → C ⊗ C (where C ⊗ C is the C*-algebra tensor product - the completion of the algebraic tensor product of C and C) such that for all i, j we have:
- There exists a linear antimultiplicative map κ: C0 → C0 (the coinverse) such that κ(κ(v*)*) = v for all v ∈ C0 and
where I is the identity element of C. Since κ is antimultiplicative, then κ(vw) = κ(w) κ(v) for all v, w in C0.
As a consequence of continuity, the comultiplication on C is coassociative.
In general, C is not a bialgebra, and C0 is a Hopf *-algebra.
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.
Representations
A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra A is a square matrix with entries in A (so v belongs to M(n, A)) such that
for all i, j and ε(vij) = δij for all i, j). Furthermore, a representation v, is called unitary if the matrix for v is unitary (or equivalently, if κ(vij) = v*ij for all i, j).
Example
An example of a compact matrix quantum group is SUμ(2), where the parameter μ is a positive real number. So SUμ(2) = (C(SUμ(2)), u), where C(SUμ(2)) is the C*-algebra generated by α and γ, subject to
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 that u is a representation, but not a unitary representation. u is equivalent to the unitary representation
Equivalently, SUμ(2) = (C(SUμ(2)), w), where C(SUμ(2)) is the C*-algebra generated by α and β, subject to
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 that w is a unitary representation. The realizations can be identified by equating .
When μ = 1, then SUμ(2) is equal to the algebra C(SU(2)) of functions on the concrete compact group SU(2).
Bicrossproduct quantum groups
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
See also
Notes
- Bibcode:1994hep.th...12237S
- ^ Andruskiewitsch, Schneider: Pointed Hopf algebras, New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.
- ^ Heckenberger: Nichols algebras of diagonal type and arithmetic root systems, Habilitation thesis 2005.
- ^ Heckenberger, Schneider: Root system and Weyl gruppoid for Nichols algebras, 2008.
- ^ Heckenberger, Schneider: Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl grupoid, 2009.
References
- Grensing, Gerhard (2013). Structural Aspects of Quantum Field Theory and Noncommutative Geometry. World Scientific. ISBN 978-981-4472-69-2.
- arXiv:math-ph/0105002.
- Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, MR 1321145
- Lusztig, George (2010) [1993]. Introduction to Quantum Groups. Cambridge, MA: Birkhäuser. ISBN 978-0-817-64716-2.
- Majid, Shahn (2002), A quantum groups primer, London Mathematical Society Lecture Note Series, vol. 292, Cambridge University Press, MR 1904789
- Majid, Shahn (January 2006), "What Is...a Quantum Group?" (PDF), Notices of the American Mathematical Society, 53 (1): 30–31, retrieved 2008-01-16
- Podles, P.; Muller, E. (1998), "Introduction to quantum groups", Reviews in Mathematical Physics, 10 (4): 511–551, S2CID 2596718
- Shnider, Steven; Sternberg, Shlomo (1993). Quantum groups: From coalgebras to Drinfeld algebras. Graduate Texts in Mathematical Physics. Vol. 2. Cambridge, MA: International Press.
- MR 2294803