Algebra of Deformed Differential Operators and Induced Integrable Toda Field Theory

We build in this paper the algebra of q-deformed pseudo-differential operators, shown to be an essential step toward setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy, namely the q-KdV and q-Boussinesq integrable systems. We also present the q-generalization of the conformal transformations of the currents u _n ,n 2, and discuss the primary condition of the fields W _n , n 2, by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented.     

A Coset-Type Construction for the Deformed Virasoro Algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U_q(sl₂). A similar construction is proposed for the elliptic algebra A_q,p(sl₂).     

Two-Parameter Deformed SUSY Algebra for Fibonacci Oscillators

We construct a two-parameter deformed SUSY algebra for the system of n ordinary fermions and n(q ₁,q ₂)-deformed bosons called Fibonacci oscillators with -symmetry. We then analyze the Fock space representation of the algebra constructed. We obtain the total deformed Hamiltonian and the energy levels together with their degeneracies for the system. We also consider some physical applications of the Fibonacci oscillators with -symmetry, and discuss the main reasons to consider two distinct deformation parameters.     

Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald- Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced; the combinatorial formulas for them are given.     

A Two-Parameter Deformed N = 2 SUSY Algebra

A two-parameter deformed N = 2 SUSY algebra is constructed by using the q-deformed bosonic and fermionic Newton oscillator algebras. The Fock space representation of the (q ₁,q ₂)-deformed N = 2 SUSY algebra is analyzed. The comparison between the algebra constructed and earlier versions of deformed N = 2 SUSY algebras is also made.     

Deformed Multiconformal Algebra and Its q-Operator Product Expansion

The q-deformed multiconformal algebra isderived. The gl_q(n)-covariant oscillatorrealization is given in the centerless case. Theq-operator product expansion realization is alsogiven.     

Nonlinear Deformed Algebra Realized in a Physical System

We show that nonlinear deformed algebra can exist in a physical system with Poschl-Teller potential. Due to this algebra, the eigenvalue problem of the system can be exactly solved by operator method. The raising and lowering operators satisfying this algebra are constructed. And the physical meaning of two deforming functions involving in this algebra is given. In addition, the SU(1,1) symmetry is exhibited in such a system by the operator method.     

Deformed C λ-Extended Heisenberg Algebra in Noncommutative Phase-Space

We construct a deformed C _λ-extended Heisenberg algebra in two-dimensional space using noncommuting coordinates which close an algebra depends on statistical parameter characterizing exotic particles. The obtained symmetry is nothing but an exotic particles algebra interpolating between bosonic and deformed fermionic algebras. PACS numbers: 03.65.Fd, 02.40.Gh, 05.30.Pr     

Whittaker Vector of Deformed Virasoro Algebra and Macdonald Symmetric Functions

We give a proof of Awata and Yamada’s conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof, we fully use currents appearing in the Fock representation of Ding–Iohara–Miki quantum algebra.     

q-Phase Operators of Finite Dimensional Two-Mode q-Oscillator Algebra

The q-phase operators are constructed for two-mode q-oscillators in a finite dimensional Hilbert space. It is shown that the q-coherent states for two-mode q-oscillators are not minimum uncertainty states.     

Two-Parameter Deformed Quantum Algebra SUq,s(4)

The commutation relations of thegenerators of the two-parameter deformed quantum algebra SU_q,s(4) are given and irreducible q,s-tensor operators of rank (1/2) of the quantum algebra SU_q,s(2) are constructed.     

Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra

We define cylindric versions of skew Macdonald functions P _λ/μ (q, t) for the special cases q = 0 or t = 0. Fixing two integers n > 2 and k > 0 we shift the skew diagram λ/μ, viewed as a subset of the two-dimensional integer lattice, by the period vector (n, ?k). Imposing a periodicity condition one defines cylindric skew tableaux and associated weight functions. The resulting weighted sums over these cylindric tableaux are symmetric functions. They appear in the coproduct of a commutative Frobenius algebra which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a ${U_{q}\widehat{\mathfrak{sl}}(n)}$ Kirillov-Reshetikhin module. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig’s canonical basis. In the limit q = t = 0, this Frobenius algebra is isomorphic to the ${\widehat{\mathfrak{sl}}(n)}$ Verlinde algebra at level k, i.e. the structure constants become the ${\widehat{\mathfrak{sl}}(n)_{k}}$ Wess-Zumino-Novikov-Witten fusion coefficients. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions discussed here arise as partition functions of so-called vertex models obtained from solutions to the Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.     

双参数变形多模玻色算符的代数结构及其应用(Ⅰ)

构造了两个满足量子Heisenberg-Weyl代数的双参数变形多模玻色算符,研究了它们的代数结构,作为应用,给出了双参数变形多模量子群SU(2)_q,s和SU/(1,1)_q,s的Holstein-Primakoff实现,并分别构造了这两个玻色算符二次幂的本征态,证明了它们的完备性.     

Enhancement non Classical Properties of a Pair of Qubit via Deformed Operators

The present paper is devoted to consider the problem of the interaction between a pair of entangled qubits and a multiphotons cavity mode. The deformed operators are involved in the Hamiltonian model which represents such system. The exact solution of the wave function is obtained and the density matrix is contracted. For two different types of Bell states, the purity as well as the fidelity of the system are investigated. In our computational program we have used different values of the photon numbers and the deformity parameter to investigate the robustness of these entangled states. It is shown that, for small values of the deformity parameter the purity and the fidelity of the travel state are improved. On the other hand, in the absence of the deformation parameter the number of photons inside the cavity improve the stability behavior for the purity and the fidelity.     

Renormalized Traces and Cocycles on the Algebra of S 1-Pseudo-differential Operators

Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators $Cl(S^1,\mathbb{C}^n)Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators acting on . We first show that the Schwinger functional associated to the Dirac operator is a cocycle on , and not only on a restricted algebra Then, we investigate two bilinear functionals and , which satisfies We show that and are two cocycles in , and and have the same nonvanishing cohomology class. We finaly calculate on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols. By this last computation, we recover the Virasoro cocyle and the K?hler form of the loop group. Mathematics Subject Classification (1991). 47G30, 47N50     

Clebsch-Gordan Coefficients for the Two-Parameter Deformed Quantum Algebra SU (1,1)q,s (I)

The Bargmann representations in the tensor product space of the irreducible representations for the two-parameter deformed quantum algebra SU(1,1)_q,s corresponding to the positive discrete series (a) are in trod uced, and the corresponding Bargmann expressions for the bases of irreps, the coherent state and the operators are also derived. The Clebsch-Gordan coefficients (CGC) for the two-parameter deformed quan tum algebra SU(1,1)_q,s corresponding to the positive discrete series (a) are obtained.     

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.