Deformed Quantum Algebras Research Articles

Quantum polynomials from deformed quantum algebras: Probability distributions, generating functions and difference equations

In this paper, we provide a novel generalization of quantum orthogonal polynomials from [Formula: see text]-deformed quantum algebras introduced in earlier works. We construct related quantum Jacobi polynomials and their probability distribution, factorial moments, recurrence relation, and governing difference equation. Surprisingly, these polynomials obey non-conventional recurrence relations. Particular cases of generalized quantum little Legendre, little Laguerre, Laguerre, Bessel, Rogers–Szegö, Stieltjes–Wigert and Kemp binomial polynomials are derived and discussed.

Generalized Heisenberg–Virasoro algebra and matrix models from quantum algebra

In this paper, we construct the Heisenberg–Virasoro algebra in the framework of the R(p,q)-deformed quantum algebras. Moreover, the R(p,q)-Heisenberg–Witt n-algebras is also investigated. Furthermore, we generalize the notion of the elliptic Hermitian matrix models. We use the constraints to evaluate the R(p,q)-differential operators of the Virasoro algebra and generalize it to higher order differential operators. Particular cases corresponding to quantum algebras existing in the literature are deduced.

ℛ(p,q)-deformed super Virasoro n-algebras

In this paper, we construct the super Witt algebra and super Virasoro algebra in the framework of the [Formula: see text]-deformed quantum algebras. Moreover, we perform the super [Formula: see text]-deformed Witt [Formula: see text]-algebra, the [Formula: see text]-deformed Virasoro [Formula: see text]-algebra and discuss the super [Formula: see text]-Virasoro [Formula: see text]-algebra ([Formula: see text] even). Besides, we define and construct another super [Formula: see text]-deformed Witt [Formula: see text]-algebra and study a toy model for the super [Formula: see text]-Virasoro constraints. Relevant particular cases induced from the quantum algebras known in the literature are deduced from the formalism developed.

Generalized Witt, Witt n-algebras, Virasoro algebras and KdV equations induced from ℛ(p,q)-deformed quantum algebras

We perform generalizations of Witt and Virasoro algebras, and derive the corresponding Korteweg–de Vries equations from known [Formula: see text]-deformed quantum algebras previously introduced in J. Math. Phys. 51 (2010) 063518. Related relevant properties are investigated and discussed. Besides, we construct the [Formula: see text]-deformed Witt [Formula: see text]-algebra, and determine the Virasoro constraints for a toy model, which play an important role in the study of matrix models. Finally, as a matter of illustration, explicit results are provided for the main particular deformed quantum algebras known in the literature.

Drinfeld double of deformed quantum algebras

We provide a deformation, fβ, of Lusztig algebra f. Various quantum algebras in literatures, including half parts of two-parameter quantum algebras, quantum superalgebras, and multi-parameter quantum algebras/superalgebras, are all specializations of fβ. Moreover, fβ is isomorphic to Lusztig algebra f up to a twist. As a consequence, half parts of those quantum algebras are isomorphic to Lusztig algebra f over a big enough ground field up to certain twists. We further construct the entire algebra Uβ,ξ by Drinfeld double construction. As special cases, above quantum algebras all admit a Drinfeld double construction under certain assumptions.

Quantum theory of the generalised uncertainty principle

We extend significantly previous works on the Hilbert space representations of the Generalized Uncertainty Principle (GUP) in 3+1 dimensions of the form $[X_i,P_j] = i F_{ij}$ where $ F_{ij} = f(P^2) \delta_{ij} + g(P^2) P_i P_j $ for any functions $f$. However, we restrict our study to the case of commuting $X$'s. We focus in particular on the symmetries of the theory, and the minimal length that emerge in some cases. We first show that, at the algebraic level, there exists an unambiguous mapping between the GUP with a deformed quantum algebra and a quadratic Hamiltonian into a standard, Heisenberg algebra of operators and an aquadratic Hamiltonian, provided the boost sector of the symmetries is modified accordingly. The theory can also be mapped to a completely standard Quantum Mechanics with standard symmetries, but with momentum dependent position operators. Next, we investigate the Hilbert space representations of these algebraically equivalent models, and focus, specifically on whether they exhibit a minimal length. We carry the functional analysis of the various operators involved, and show that the appearance of a minimal length critically depends on the relationship between the generators of translations and the physical momenta. In particular, because this relationship is preserved by the algebraic mapping presented in this paper, when a minimal length is present in the standard GUP, it is also present in the corresponding Aquadratic Hamiltonian formulation, despite the perfectly standard algebra of this model. In general, a minimal length requires bounded generators of translations, i.e. a specific kind of quantization of space, and this depends on the precise shape of the function $f$ defined previously. This result provides an elegant and unambiguous classification of which universal quantum gravity corrections lead to the emergence of a minimal length.

Primordial fluctuations from deformed quantum algebras

We study the implications of deformed quantum algebras for the generation of primordial perturbations from slow-roll inflation. Specifically, we assume that the quantum commutator of the inflaton's amplitude and momentum in Fourier space gets modified at energies above some threshold M⋆. We show that when the commutator is modified to be a function of the momentum only, the problem of solving for the post-inflationary spectrum of fluctuations is formally equivalent to solving a one-dimensional Schr"odinger equation with a time dependent potential. Depending on the class of modification, we find results either close to or significantly different from nearly scale invariant spectra. For the former case, the power spectrum is characterized by step-like behaviour at some pivot scale, where the magnitude of the jump is \U0001d4aa(H2/M⋆2). (H is the inflationary Hubble parameter.) We use our calculated power spectra to generate predictions for the cosmic microwave background and baryon acoustic oscillations, hence demonstrating that certain types of deformations are incompatible with current observations.

( R , p , q ) -deformed quantum algebras: Coherent states and special functions

We provide with a generalization of well known (p,q)-deformed Heisenberg algebras, called (R,p,q)-deformed quantum algebras, and study the corresponding (R,p,q)-series. A general formulation of the binomial theorem is given. Special functions are obtained as limit cases. This work well prolongs a previous work by Odzijewicz [Commun. Math. Phys. 192, 183 (1998)]. Known results in the literature are recovered.

Yang-Baxter algebra and generation of quantum integrable models

An operator deformed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its $q \to 1$ limit appear to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect.

Formula omitted]-Deformed quantum Lie algebras

Attention is focused on q -deformed quantum algebras with physical importance, i.e. U q ( su 2 ) , U q ( so 4 ) and q -deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry algebras in a consistent framework which will serve as starting point for representation theoretic investigations in physics, especially quantum field theory. In each case considerations start from a realization of symmetry generators within the differential algebra. Formulae for coproducts and antipodes on symmetry generators are listed. The action of symmetry generators in terms of their Hopf structure is taken as the q -analog of classical commutators and written out explicitly. Spinor and vector representations of symmetry generators are calculated. A review of the commutation relations between symmetry generators and components of a spinor or vector operator is given. Relations for the corresponding quantum Lie algebras are computed. Their Casimir operators are written down in a form similar to that for the undeformed case.

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

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 negative discrete series (b) are introduced, 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 quantum algebra SU(1,1)q,s corresponding to the negative discrete series (b) are obtained.

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

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

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 introduced, 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 quantum algebra SU(1,1)q,s corresponding to the positive discrete series (a) are obtained.

Clebsch–Gordan Coefficients for the Two-Parameter Deformed Quantum Algebra SU(2)q,s

The Bargmann representations in the tensor product space of the irreducible representations for the two-parameter deformed quantum algebra SU(2)q,s are introduced. The Bargmann expressions for the bases of irreps, the coherent state and the operators are also derived. In terms of those expressions, the calculation of the Clebsch–Gordan coefficients (CGC) for the two-parameter deformed quantum algebra SU(2)q,s becomes very easy.