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arXiv1997

Representations of the q-deformed algebra $U'_q(so_{2,2})$

The main aim of this paper is to give classes of irreducible infinite dimensional representations and of irreducible $*$-representations of the q-deformed algebra $U'_q(so_{2,2})$ which is a real form of the non-standard deformation $U'_q(so_4)$ of the universal enveloping algebra $U(so(4,C))$. These representations are described by two complex parameters and are obtained by "analytical continuation" of the irreducible finite dimensional representations of the algebra $U'_q(so_4)$ in the basis corresponding to the reduction from $U'_q(so_4)$ to $U(so_2\oplus so_2)$.

q-alg↗
arXiv1997

Hochschild cohomology and characteristic classes for star-products

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation parameter is seen to be closely related to the characteristic class of the quantization. A fundamental role in the analysis is played by ``quantum Liouville operators,'' which rescale the deformation parameter in the same way in which Liouville vector fields scale the Poisson structure (or the units of action). Several examples are given.

q-alg↗
arXiv1997

Lie Algebras in Fock Space

A catalogue of explicit realizations of representations of (super) Lie algebras and quantum algebras in Fock space is presented.

q-alg↗
arXiv1997

Boundary solutions of the quantum Yang-Baxter equation and solutions in three dimensions

Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ``modified'' qYB equation, the latter being analogous to the modified classical Yang-Baxter (cYB) equation. We construct, for a large class of solutions $r$ to the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary solutions to the qYB equation of the form $I+tr+ t^2r_{2} + ...$. In the last section we list and give quantizations for all classical r-matrices in $sl(3) \wedge sl(3)$.

q-alg↗
arXiv1997

Biorthogonal polynomials associated with reflection groups and a formula of Macdonald

Dunkl operators are differential-difference operators on $\b R^N$ which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we introduce two systems of biorthogonal polynomials with respect to Dunkl's Gaussian distributions in a quite canonical way. These systems, called Appell systems, admit many properties known from classical Hermite polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian distributions. In particular, these polynomials lead to a new proof of a generalized formula of Macdonald due to Dunkl. The ideas for this paper are taken from recent works on non-Gaussian white noise analysis and from the umbral calculus.

q-alg↗
arXiv1998

Exotic Differential Operators on Complex Minimal Nilpotent Orbits

Let O be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action $C^*$ on g. The algebra of differential operators on O is D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \cite{jos} and \cite{bkHam} for some results on the geometry and quantization of O. We construct an explicit subspace $A_{-1}\subset D(O)$ of commuting differential operators which are Euler homogeneous of degree -1. The space $A_{-1}$ is finite-dimensional, g-stable and carries the adjoint representation. $A_{-1}$ consists of (for $g \neq sp(2n,C)$) non-obvious order 4 differential operators obtained by quantizing symbols we obtained previously. These operators are "exotic" in that there is (apparently) no geometric or algebraic theory which explains them. The algebra generated by $A_{-1}$ is a maximal commutative subalgebra A of D(X). We find a G-equivariant algebra isomorphism R(O) to A, $f\mapsto D_f$, such that the formula $(f|g)=({constant term of}D_{\bar{g}} f)$ defines a positive-definite Hermitian inner product on R(O). We will use these operators $D_f$ to quantize O in a subsequent paper.

q-alg↗
arXiv1997

FRT Quantization Theory for the Nonsemisimple Cayley-Klein Groups

The quantization theory of the simple Lie groups and algebras was developed by Faddeev-Reshetikhin-Takhtadjan (FRT). In group theory there is a remarkable set of groups, namely the motion groups of n-dimensional spaces of constant curvature or the orthogonal Cayley-Klein (CK) groups. In some sense the CK groups are in the nearest neighborhood with the simple ones. The well known groups of physical interest such as Euclidean E(n), Poincare P(n), Galileian G(n) and other nonsemisimple groups are in the set of CK groups. But many standart algebraical costructions are not suitable for the nonsemisimple groups and algebras, in particular Killing form is degenerate, Cartan matrix do not exist. Nevertheless it is possible to describe and to quantize all CK groups and algebras, as it was made for the simple ones. The principal proposal is to consider CK groups as the groups over an associative algebra $D$ with nilpotent commutative generators and the corresponding quantum CK groups as the algebra of noncommutative functions over $D$.

q-alg↗
arXiv1998

Quasi-Hopf twistors for elliptic quantum groups

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U_q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U_q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. This construction entails that, for generic values of the deformation parameters, representation theory for U_q(g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebra A_{q,p}(^sl_2).

q-alg↗
arXiv1997

Theory of linear G-difference equations

We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations,symmetries of an equation are module endomorphisms and conserved structures are invariants in the tensor algebra of the given equation. We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underluing set. We relate our notion of difference equations and solutions to systems of classical difference equations and their solutions and show that our notions include these as a special case.

q-alg↗
arXiv1997

Centralizer construction for twisted Yangians

For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type B, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal enveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be naturally projected onto the $(n-1)$th one, so that one can form the projective limit of the centralizer algebras as $n\to\infty$ with $m$ fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by $A_m$. We explicitly construct an algebra isomorphism $A_m=Z\otimes Y_m$, where $Z$ is a commutative algebra and $Y_m$ is the so-called twisted Yangian associated to the rank $m$ classical Lie algebra of type B, C, or D. The algebra $Z$ may be viewed as the algebra of virtual Laplace operators; it is isomorphic to the algebra of polynomials with countably many indeterminates. The twisted Yangian $Y_m$ (and hence the algebra $A_m$) can be described in terms of a system of generators with quadratic and linear defining relations which are conveniently presented in R-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case by the second author.

q-alg↗
arXiv1995

On the Problem of Chaos Conservation in Quantum Physics

We develop a new method of constructing a large N asymptotic series in powers of $N^{-1/2}$ for the function of N arguments which is a solution to the Cauchy problem for the equation of a special type. Many-particle Wigner, Schrödinger and Liouville equations for a system of a large number of particles are of this type, when the external potential is of order O(1), while the coefficient of the particle interaction potential is 1/N; the potentials can be arbitrary smooth bounded functions. We apply this method to equations for N-particle states corresponding to the N-th tensor power of an abstract Hamiltonian algebra of observables. In particular, we show for the case of multiparticle Schrödinger-like equations that the property of N-particle wave function to be approximately equal at large N to the product of one-particle wave functions does not conserve under time evolution, while the same property for the correlation functions of the finite order is known to conserve(such hypothesis being the quantum analog of the chaos conservation hypothesis put forward by M.Kac in 1956 was proved by the analysis of the BBGKY-like hierarchy of equations). In order to find a leading asymptotics for the N-particle wave function, one should use not only the solution to the well- known Hartree equation being derivable from the BBGKY approach but also the solution to another (Riccati-type) equation presented in this paper. We also consider another interesting case when one adds to the N-particle system under consideration one more particle interacting with the system with the coefficient of the interaction potential of order O(1).It happens that in this case one should investigate not a single Hartree-like equation but a set of such equations, and the chaos will not conserve even for the correlators.

quant-ph↗
arXiv1997

A Modification of Nambu's Mechanics

The Poisson, contact and Nambu brackets define algebraic structures on $C^{\infty}(M)$ satisfying the Jacobi identity or its generalization. The automorphism groups of these brackets are the symplectic, contact and volume preserving diffeomorphism groups. We introduce a modification of the Nambu bracket, which defines an evolution equation generating the whole diffeomorphism group. The relation between the modified and usual Nambu brackets is similar to the relation between the Poisson and contact structures. We briefly discuss the problem of quantization of the modified bracket.

quant-ph↗
arXiv1997

Is the Deformation Parameter in q-Rotor Model Really Phenomenological ?

We cast the $q$-rotor in the framework of Barnett-Pegg theory for rotation angle, whose underlying algebra is $SU_q(2)$. A new method to fix the deformation parameter from the theory is suggested. We test our ideas by fitting rotational spectra in deformed even-even and superdeformed nuclei. The results are in good agreement with the previous phenomenological applications of $q$-rotor model.

quant-ph↗
arXiv1997

Supersymmetric Construction of Exactly Solvable Potentials and Non-Linear Algebras

Using algebraic tools of supersymmetric quantum mechanics we construct classes of conditionally exactly solvable potentials being the supersymmetric partners of the linear or radial harmonic oscillator. With the help of the raising and lowering operators of these harmonic oscillators and the SUSY operators we construct ladder operators for these new conditionally solvable systems. It is found that these ladder operators together with the Hamilton operator form a non-linear algebra which is of quadratic and cubic type for the SUSY partners of the linear and radial harmonic oscillator, respectively.

quant-ph↗
arXiv1997

The k-fermions as objects interpolating between fermions and bosons

Operators, refered to as k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of two noncommuting quon algebras. The deformation parameters for these quon algebras are roots of unity connected to an integer k. The case k=2 corresponds to fermions and the limiting case k going to infinity to bosons. Generalized coherent states and supercoherent states are investigated. The Dirac quantum phase operator and the Fairlie-Fletcher-Zachos algebra are also considered.

quant-ph↗