Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -?² plus a quantum deformation of order ?⁴ and ?⁶. The systems inside the classes are transformed using Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.     

Spectral stochastic processes arising in quantum mechanical models with a non-L 2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non-L ² ground state. As a prototype, the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal Abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite-volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite-volume measures.Supported by DFG, Nr. Al 374/1-2     

Convolution Inequalities for the Boltzmann Collision Operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L ^p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L ² case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some L^s_weak{L^{s}_{weak}} or L ^s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

R-Matrix Quantization of the Elliptic Ruijsenaars–Schneider Model

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and ?r-matrices satisfying a closed system of equations. The corresponding quantum R and ?R-matrices are found as solutions to quantum analogs of these equations. We present the quantum L-operator algebra and show that the system of equations on R and ?R arises as the compatibility condition for this algebra. It turns out that the R-matrix is twist-equivalent to the Felder elliptic R ^F -matrix with ?R playing the role of the twist. The simplest representation of the quantum L-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum L-operator algebra to the fundamental relation RLL=LLR with Belavin's elliptic R matrix is established. As a byproduct of our construction, we find a new N-parameter elliptic solution to the classical Yang-Baxter equation. Received: 17 March 1997 / Accepted: 8 July 1997     

Duality for Coloured Quantum Groups

Duality between the coloured quantum group and the coloured quantum algebra corresponding to GL(2) is established. The coloured L ^± functionals are constructed and the dual algebra is derived explicitly. These functionals are then employed to give a coloured generalisation of the differential calculus on quantum GL(2) within the framework of the R-matrix approach.     

Physical states and BRST operators for higher-spin <Emphasis Type="Italic">W</Emphasis> strings

In this paper, we mainly investigate the W _2,s ^M ⊗ W _2,s ^L system, in which the matter and the Liouville subsystems generate the W _2,s ^M and W _2,s ^L algebras, respectively. We first give a brief discussion of the physical states for the corresponding W strings. The lower states are given by freezing the spin-2 and spin-s currents. Then, introducing two pairs of ghost-like fields, we give the realizations of the W _1,2,s algebras. Based on these linear realizations, the BRST operators for the W _2,s algebras are obtained. Finally, we construct new BRST charges of the Liouville system for the W _2,s ^L strings at the specific values of the central charges c: for the W _2,3^L algebra, c=−24 for the W _2,4^L algebra and for the W _2,6^L algebra, at which the corresponding W _2,s ^L algebras are singular.     

Construction of cyclic representations of quantum algebras at q p =1 from their regular representations

Cyclic representations of maximal dimension of the quantum algebra U _q L associated with any finite-dimensional simple Lie algebra L are studied from its regular representation at q ^p =1, which is proved to be a quotient module of itself as a left module with respect to some submodules. The general theory is given after an instructive example U _q sl(2) is studied. Another explicit example U _q sl(3) is also presented.This work is supported in part by the National Natural Science Foundation of China. Author Fu is also supported by the Jilin Provincial Science and Technology Foundation of China     

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.     

Expanding spherically symmetric models without shear

The integrability properties of the field equationL _xx =F(x)L ² of a spherically symmetric shear-free fluid are investigated. A first integral, subject to an integrability condition onF(x), is found, giving a new class of solutions which contains the solutions of Stephani and Srivastava as special cases. The integrability condition onF(x) is reduced to a quadrature which is expressible in terms of elliptic integrals in general. There are three classes of solution and in general the solution ofL _xx =F(x)L ² can only be written in parametric form. The case for whichF=F(x) can be explicitly given corresponds to the solution of Stephani. A Lie analysis ofL _xx =F(x)L ² is also performed. If a constant vanishes, then the solutions of Kustaanheimo and Qvist and of this paper are regained. For 0 we reduce the problem to a simpler, autonomous equation. The applicability of the Painlevé analysis is also briefly considered.     

Hypercontractivity on the <Emphasis Type="Italic">q</Emphasis>-Araki-Woods Algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the q-Ornstein-Uhlenbeck semigroup on the q-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the q-Araki-Woods algebras. We show that hypercontractivity from L ^p to L ² can occur if and only if the generator of the deformation is bounded.     

Algebra of Screening Operators for the Deformed W n Algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed W _n algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W ₃ algebra. Received: 3 March 1997 / Accepted: 20 May 1997     

Trace Functionals for a Class of Pseudo-Differential Operators in R n

In this paper we define trace functionals on the algebra of pseudo-differential operators with cone-shaped exits to infinity. Furthermore, we improve the Weyl formula on the asymptotic distribution of eigenvalues and make use of it in order to establish inclusion relations between the interpolation normed ideals of compact operators in L ²(R ⁿ ) and the above operator classes.     

Quantum algebra structure of certain Jackson integrals

Theq-difference system satisfied by Jackson integrals with a configuration ofA-type root system is studied. We explicitly construct some linear combination of Jackson integrals, which satisfies the quantum Knizhnik-Zamolodchikov equation for the 2-point correlation function ofq-vertex operators, introduced by Frenkel and Reshetikhin, for the quantum affine algebra . The expression of integrands for then-point case is conjectured, and a set of linear relations for the corresponding Jackson integrals is proved.     

Linking numbers,contacts, and mutual inductances of a random set of closed curves

We establish here a new, general result of integral geometry, concerning closed rigid curves of arbitrary shapes inE ³ and their linking numbersI. It generalizes by a different method, the interesting integral property ofI ² found recently by Pohl and extended by des Cloizeaux and Ball, for two curves. We considern closed curves linked successively to each other and forming a ring. The cyclic product of their linking numbers is integrated over the group of rigid motions of the curves. This integral is shown to factorize over a special algebra of characteristic functions. Each curve possesses two such intrinsic functions. The same algebra is shown to describe a larger class of integral geometry properties: a new theorem is established for a family of displacement integrals involving linking numbers, contact angles, and mutual inductances of the set ofn curves.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

Algebraic Reduction of the Ising Model

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.     

Hypercontractivity for Semigroups of Unital Qubit Channels

Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form ${e^{-t_1 H_1}\otimes \cdots \otimes e^{- t_n H_n}}$ to be a contraction from L ^p to L ^q , where L ^p is the algebra of 2 ⁿ -dimensional matrices equipped with the normalized Schatten norm, and each generator H _j is a self-adjoint positive semidefinite operator on the algebra of 2-dimensional matrices. As a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels.     

Vanishing of the Kontsevich Integrals of the Wheels

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. These integrals appear in the approach to the Duflo formula via the formality theorem. The result means that for any finite-dimensional Lie algebra g, and for invariant polynomials f, g [S ^·(g)]^g one has f · g = f * g, where * is the Kontsevich star product, corresponding to the Kirillov–Poisson structure on g^*. We deduce this theorem form the result contained in math.QA/0010321 on the deformation quantization with traces.