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1.
S. Lievens N. I. Stoilov J. Van der Jeugt 《Communications in Mathematical Physics》2008,281(3):805-826
It is known that the defining relations of the orthosymplectic Lie superalgebra are equivalent to the defining (triple) relations of n pairs of paraboson operators . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of . Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the
present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra of plays a crucial role. Our results also lead to character formulas for these infinite-dimensional representations. Furthermore, by considering the branching , we find explicit infinite-dimensional unitary irreducible lowest weight representations of and their characters.
NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity
Attraction Poles Programme (Belgian State – Belgian Science Policy).
An erratum to this article can be found at 相似文献
2.
S. M. Khoroshkin I. I. Pop M. E. Samsonov A. A. Stolin V. N. Tolstoy 《Communications in Mathematical Physics》2008,282(3):625-662
We study classical twists of Lie bialgebra structures on the polynomial current algebra , where is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric
solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of . We give the complete classification of quasi-trigonometric r-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root).
We quantize solutions corresponding to the first root of . 相似文献
3.
We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular
Poisson submanifolds by intersecting certain group orbits. Examples include all varieties of Lagrangian subalgebras of reductive quadratic Lie algebras with Poisson structures defined by Lagrangian splittings of . In the special case of , where is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on defined by arbitrary Lagrangian splittings of . Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin–Drinfeld splittings as special
cases. 相似文献
4.
In this paper we introduce Baxter integral -operators for finite-dimensional Lie algebras and . Whittaker functions corresponding to these algebras are eigenfunctions of the -operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the
manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions,
which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental
integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump
and Bump conjectures for G = GL(ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter -operator acting on Whittaker functions with local Archimedean L-factors. The Baxter -operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter
operator in the spherical Hecke algebra , K being a maximal compact subgroup of G. Finally we stress an analogy between -operators and certain elements of the non-Archimedean Hecke algebra . 相似文献
5.
For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f
0, f
1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f. 相似文献
6.
Quantum Conjugacy Classes of Simple Matrix Groups 总被引:1,自引:0,他引:1
A. Mudrov 《Communications in Mathematical Physics》2007,272(3):635-660
Let G be a simple complex classical group and its Lie algebra. Let be the Drinfeld-Jimbo quantization of the universal enveloping algebra . We construct an explicit -equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers.
Dedicated to the memory of Joseph Donin
This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany,
the Excellency Center “Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, by the
EPSRC grant C511166, and by the RFBR grant no. 06-01-00451. 相似文献
7.
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients
. In the physical regime of a small inelasticity (that is for some constructive ) we prove uniqueness of the self-similar profile for given values of the restitution coefficient , the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation).
Moreover, if the initial datum lies in , and under some smallness condition on depending on the mass, energy and norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar
solution (the so-called homogeneous cooling state).
These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding
results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling,
and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic
limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the “quasi-elastic
self-similar temperature” and the rate of convergence towards self-similarity at first order in terms of (1−α), are obtained
from our study.
These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic
hard spheres with small inelasticity. 相似文献
8.
A trajectory attractor is constructed for the 2D Euler system containing an additional dissipation term −ru, r > 0, with periodic boundary conditions. The corresponding dissipative 2D Navier-Stokes system with the same term −ru and with viscosity v > 0 also has a trajectory attractor, . Such systems model large-scale geophysical processes in atmosphere and ocean (see [1]). We prove that → as v → 0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit of the trajectory attractors as v → 0+. We prove that is a connected invariant subset of . The connectedness problem for the trajectory attractor by itself remains open.
Dedicated to the memory of Leonid Volevich
Partially supported by the Russian Foundation for Basic Research (projects no 08-01-00784 and 07-01-00500). The first author
has been partially supported by a research grant from the Caprio Foundation, Landau Network-Cento Volta. 相似文献
9.
Asao Arai 《Letters in Mathematical Physics》2008,85(1):15-25
Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e−itH
D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation .
This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science
(JSPS). 相似文献
10.
For a (co)monad T
l
on a category , an object X in , and a functor , there is a (co)simplex in . The aim of this paper is to find criteria for para-(co)cyclicity of Z
*. Our construction is built on a distributive law of T
l
with a second (co)monad T
r
on , a natural transformation , and a morphism in . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation
that a (co)ring T over an algebra R determines a distributive law of two (co)monads and on the category of R-bimodules. The functor Π can be chosen such that is the cyclic R-module tensor product. A natural transformation is given by the flip map and a morphism is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called ×
R
-Hopf algebras, is introduced. In the particular example when T is a module coring of a ×
R
-Hopf algebra and X is a stable anti-Yetter-Drinfel’d -module, the para-cyclic object Z
* is shown to project to a cyclic structure on . For a -Galois extension , a stable anti-Yetter-Drinfel’d -module T
S
is constructed, such that the cyclic objects and are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild
and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the
group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies
of a groupoid. The latter extends results of Burghelea on cyclic homology of groups. 相似文献
11.
We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping
class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first
universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion .
L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311. 相似文献
12.
13.
We define the twisted loop Lie algebra of a finite dimensional Lie algebra
as the Fréchet space of all twisted periodic smooth mappings from
to
. Here the Lie algebra operation is continuous. We call such Lie algebras Fréchet Lie algebras. We introduce the notion of an integrable
-gradation of a Fréchet Lie algebra, and find all inequivalent integrable
-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.On leave of absence from the Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia. 相似文献
14.
Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This
function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc., and it can also be interpreted
as correlation functions on integrable -modules of level one. Such -correlation functions at higher levels were then calculated by Cheng and Wang.
In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of of type B, C, D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B, C, D and some fermionic type q-identities. 相似文献
15.
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation 总被引:2,自引:0,他引:2
We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first
prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve and the set of non-characteristic points. We show that I
0 is open and that T(x) is C
1 on I
0. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for .
This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185. 相似文献
16.
Valter Moretti 《Communications in Mathematical Physics》2006,268(3):727-756
This work concerns some features of scalar QFT defined on the causal boundary
of an asymptotically flat at null infinity spacetime and based on the BMS-invariant Weyl algebra
.(a) (i) It is noticed that the natural BMS invariant pure quasifree state λ on
, recently introduced by Dappiaggi, Moretti and Pinamonti, enjoys positivity of the self-adjoint generator of u-translations with respect to every Bondi coordinate frame
on
, (
being the affine parameter of the complete null geodesics forming
and
complex coordinates on the transverse 2-sphere). This fact may be interpreted as a remnant of the spectral condition inherited from QFT in Minkowski spacetime (and it is the spectral condition for free fields when the bulk is the very Minkowski space). (ii) It is also proved that the cluster property under u-displacements is valid for every (not necessarily quasifree) pure state on
which is invariant under u displacements. (iii) It is established that there is exactly one algebraic pure quasifree state which is invariant under u-displacements (of a fixed Bondi frame) and has positive self-adjoint generator of u-displacements. It coincides with the GNS-invariant state λ. (iv) Finally it is shown that in the folium of a pure u-displacement invariant state ω (like λ but not necessarily quasifree) on
is the only state invariant under u-displacement.(b) It is proved that the theory can be formulated for spacetimes asymptotically flat at null infinity which also admit future time completion i
+ (and fulfill other requirements related with global hyperbolicity). In this case a -isomorphism ı exists - with a natural geometric meaning - which identifies the (Weyl) algebra of observables of a linear field propagating in the bulk spacetime with a sub algebra of
. Using ı a preferred state on the field algebra in the bulk spacetime is induced by the BMS-invariant state λ on
. 相似文献
17.
We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W
n
(S), and rooted self-avoiding polygons P
n
(S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P
n
(S), and W
n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and . 相似文献
18.
Using the Godement mean of positive-type functions over a groupG, we study -abelian systems {
, } of aC*-algebra
and a homomorphic mapping of a groupG into the homomorphism group of
. Consideration of the Godement mean off(g)U
g
withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) (
g
(A)) withA
and a covariant representation of the system {
, } for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over
. Finally we investigate the discrete spectrum of covariant representations of {
, } (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).On leave of absence from Istituto di Fisica G. Marconi Piazzale delle Scienze 5 — Roma. 相似文献
19.
We continue our study of the collision of two solitons for the subcritical generalized KdV equations
Solitons are solutions of the type where c
0 > 0. In [21], mainly devoted to the case f (u) = u
4, we have introduced a new framework to understand the collision of two solitons , for (0.1) in the case (or equivalently, ). In this paper, we consider the case of a general nonlinearity f (u) for which , are nonlinearly stable. In particular, since f is general and c
1 can be large, the results are not perturbations of the ones for the power case in [21].
First, we prove that the two solitons survive the collision up to a shift in their trajectory and up to a small perturbation
term whose size is explicitly controlled from above: after the collision, , where is close to c
j
(j = 1, 2). Then, we exhibit new exceptional solutions similar to multi-soliton solutions: for all , there exists a solution such that
where (j = 1, 2) and converges to 0 in a neighborhood of the solitons as .
The analysis is split in two distinct parts. For the interaction region, we extend the algebraic tools developed in [21] for
the power case, by expanding f (u) as a sum of powers plus a perturbation term. To study the solutions in large time, we rely on previous tools on asymptotic
stability in [17,22] and [18], refined in [19,20].
This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN). 相似文献
20.
David Gérard-Varet 《Communications in Mathematical Physics》2009,286(1):81-110
We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous
random process, with typical size . In the parent paper [8], we derived a homogenized boundary condition of Navier type as . We show here that for a large class of boundaries, this Navier condition provides a approximation in L
2, instead of for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this
boundary layer are deduced from a central limit theorem for dependent variables. 相似文献