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1.
In this paper we provide a general condition for the reducibility of the Reshetikhin–Turaev quantum representations of the
mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial
genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular
tensor category are reducible. In particular, for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin–Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete
decomposition into irreducible representations for high enough levels. 相似文献
2.
David Hernandez 《Communications in Mathematical Physics》2007,276(1):221-259
In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin
modules of quantum affine algebras introduced in [Cha1]). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property
for a large class of minimal affinizations in types C, D, F. As an application, the Frenkel-Mukhin algorithm [FM1] works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of [NN1] (already known for type A). The proof of the special property
is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras. 相似文献
3.
Let (, d) be a first-order differential *-calculus on a *-algebra
. We say that a pair (, F) of a *-representation of
on a dense domain
of a Hilbert space and a symmetric operator F on
gives a commutator representation of if there exists a linear mapping : L(
) such that (adb) = (a)i[F, (b) ], a, b
. Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra
has a faithful commutator representation. For a class of bicovariant *-calculi on
, there is a commutator representation such that F is the image of a central element of the quantum tangent space. If
is the Hopf *-algebra of the compact form of one of the quantum groups SL
q
(n+1), O
q
(n), Sp
q
(2n) with real trancendental q, then this commutator representation is faithful. 相似文献
4.
5.
Kevin McGerty 《Communications in Mathematical Physics》2010,296(1):89-109
We give a representation-theoretic interpretation of the Langlands character duality of [FH], and show that the “Langlands
branching multiplicities” for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For
finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules. 相似文献
6.
By constructing the q-analogue of the Heisenberg-Weyl algebra in terms of usual creation and annihilation operators of boson states in the Fock space, the boson realization method recently suggested[7-11] is generalized to obtain a class of representations of quantum group in the Fock space. The q-deformed differential realization of quantum groups proposed by Alvarez-Gaume is derived by making use of the boson realization in this paper. 相似文献
7.
Harold Steinacker 《Communications in Mathematical Physics》1998,192(3):687-706
We study irreducible unitary representations of U
q
(SO(2,1)) and U
q
(SO(2,?3)) for q a root of unity, which are finite dimensional. Among others, unitary representations corresponding to all classical one-particle
representations with integral weights are found for , with M being large enough. In the “massless” case with spin bigger than or equal to 1 in 4 dimensions, they are unitarizable only
after factoring out a subspace of “pure gauges” as classically. A truncated associative tensor product describing unitary
many-particle representations is defined for .
Received: 27 November 1996 / Accepted: 28 July 1997 相似文献
8.
Hadi Salmasian 《Communications in Mathematical Physics》2010,297(1):189-227
We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished
class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical
Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite
coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford
super Lie groups. 相似文献
9.
Groups of smooth maps from spheres to appropriate nilpotent Lie groups exhibit some peculiar properties of the unitary duals
of infinite-dimensional groups.
Received: 22 October 2000 / Accepted: 22 November 2002
Published online: 21 February 2003
RID="⋆"
ID="⋆" This research was supported in part by CONICET, FONCYT, CONICOR and UNC.
Communicated by H. Araki 相似文献
10.
Jintai Ding 《Communications in Mathematical Physics》1999,200(2):399-420
This is an extension of quantum spinor construction in [DF2]. We define quantum affine Clifford algebras based on the tensor
product of a finite dimensional representation and an infinite highest weight representation of and the solutions of q-KZ equations, construct quantum spinor representations of and explain classical and quantum boson-fermion correspondence.
Received: 22 November 1995 / Accepted: 21 July 1998 相似文献
11.
Positive Energy Representations of the Loop Groups of Non-Simply Connected Lie Groups 总被引:1,自引:1,他引:0
Valerio Toledano Laredo 《Communications in Mathematical Physics》1999,207(2):307-339
We classify and construct all irreducible positive energy representations of the loop group of a compact, connectedand simple
Lie group and show that they admit an intertwining action of Diff(S
1).
Received: 7 April 1998 / Accepted: 26 April 1999 相似文献
12.
In this letter, a new class of represen tations of the braid groups BN, Ii-1⊕ T ⊕ IN-i-1 is constructed. It is proved that those represen tations contain three kinds of irreducible representations: the trivial (identity) one, the Burau one, and a new N-dimensional one. The explicit form of the N-dimensional irreducible representation of the braid group BN is given. 相似文献
13.
For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid),
whose representation theory is essentially the same as the representation theory of the quantum group U
q
(?). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation
theory of U
q
(?), and is an algebraic structure standing behind these relations.
Received: 24 March 1998 / Accepted: 14 February 1999 相似文献
14.
The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models. 相似文献
15.
S. M. Arkhipov 《Communications in Mathematical Physics》1997,188(2):379-405
We define a new cohomology theory of associative algebras called semiinfinite cohomology in the derived categories' setting.
We investigate the case of a small quantum group u, calculate semiinfinite cohomology spaces of the trivial u-module and express them in terms of local cohomology of the nilpotent cone for the corresponding semisimple Lie algebra.
We discuss the connection between the semiinfinite homology of u and the conformal blocks' spaces.
Received: 14 October 1996 / Accepted: 25 February 1997 相似文献
16.
It has been maintained that the physical content of a model of a system is completely contained in the C∗-algebra of quasi-local observables
that is associated with the system. The reason given for this is that the unitarily inequivalent representations of
are physically equivalent. But, this view is dubious for at least two reasons. First, it is not clear why the physical content
does not extend to the elements of the von Neumann algebras that are generated by representations of
. It is shown here that although the unitarily inequivalent representations of
are physically equivalent, the extended representations are not. Second, this view detracts from special global features
of physical systems such as temperature and chemical potential by effectively relegating them to the status of fixed parameters.
It is desirable to characterize such observables theoretically as elements of the algebra that is associated with a system
rather than as parameters, and thereby give a uniform treatment to all observables. This can be accomplished by going to larger
algebras. One such algebra is the universal enveloping von Neumann algebra, which is generated by the universal representation
of
; another is the direct integral of factor representations that are associated with the set of values of the global features.
Placing interpretive significance on the von Neumann algebras mentioned earlier sheds light on the significance of unitarily
inequivalent representations of
, and it serves to show the limitations of the notion of physical equivalence. 相似文献
17.
We introduce an affinization of the quantum Kac–Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac–Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial indentity about Hall–Littlewood polynomials. 相似文献
18.
We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on 2, i.e. on the space of square summable functions on a one-dimensional lattice. 相似文献
19.
Representations of the Weyl Algebra in Quantum Geometry 总被引:2,自引:0,他引:2
Christian Fleischhack 《Communications in Mathematical Physics》2009,285(1):67-140
The Weyl algebra of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is
studied. It is shown that, in the piecewise analytic category, every regular representation of having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional
assumptions concern the dimension of the underlying analytic manifold (at least three), the finite wide triangulizability
of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms – but neither any domain properties
of the represented Weyl operators nor the requirement that the diffeomorphisms act by pull-backs. For this, the general behaviour
of C*-algebras generated by continuous functions and pull-backs of homeomorphisms, as well as the properties of stratified analytic
diffeomorphisms are studied. Additionally, the paper includes also a short and direct proof of the irreducibility of . 相似文献
20.
A simplified construction of representations is presented for the quantized enveloping algebra
q (
), with
being a simple complex Lie algebra belonging to one of the four principal series A\ell, B\ell, C\ell or D\ell. The carrier representation space is the quantized algebra of polynomials in antiholomorphic coordinate functions on the big cell of a coadjoint orbit of K where K is the compact simple Lie group with the Lie algebra
– the compact form of
. 相似文献