Para-Grassmann Star Product Calculation (original) (raw)
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Journal of Mathematical Physics, 1993
This paper continues a study of one and two variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satis ed by q-hypergeometric functions. Here we consider the quantum algebra Uq (su 2 ). We show that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the \group operators" on these representation spaces. This \local" approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. We show that the matrix elements themselves transform irreducibly under the action of the quantum algebra. We nd an alternate and simpler derivation of a q-analog, due to Groza, Kachurik and Klimyk, of the Burchnall-Chaundy formula for the product of two hypergeometric functions 2 F 1 . It is interpreted here as the expansion of the matrix elements of a \group operator" (via the exponential mapping) in a tensor product basis in terms of the matrix elements in a reduced basis.
Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations
Journal of Algebra and Its Applications, 2020
To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.
Linear Algebra and its Applications, 2020
We construct a mapping from Λ to Δ which is bijective if N > 2D and 2 : 1 if N = 2D. We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T-modules, by embedding the standard module of J q (N, D) in a bigger space that allows a U √ q (sl 2)-module structure [9]. As a byproduct we have the following: for a fixed ρ, 0 ≤ ρ ≤ D, set N = N −2ρ, D = D−ρ, and Λ ρ = {(α, β) | (α, β, ρ) ∈ Λ}. Then Λ ρ is precisely the set that parameterizes the isomorphism classes of irreducible T-modules for the Johnson scheme J(N , D) [3].
Q-boson coherent states and para-Grassmann variables for multi-particle states
Journal of Physics A: Mathematical and Theoretical, 2012
We describe coherent states and associated generalized Grassmann variables for a system of m independent q-boson modes. A resolution of unity in terms of generalized Berezin integrals leads to generalized Grassmann symbolic calculus. Formulae for operator traces are given and the thermodynamic partition function for a system of q-boson oscillators is discussed.
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arXiv (Cornell University), 2013
This paper addresses a construction of new q−Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem.
Coherent state quantization of paragrassmann algebras
2010
By using a coherent state quantization of paragrassmann variables, operators are constructed in finite Hilbert spaces. We thus obtain in a straightforward way a matrix representation of the paragrassmann algebra. This algebra of finite matrices realizes a deformed Weyl-Heisenberg algebra. The study of mean values in coherent states of some of these operators leads to interesting conclusions.
Models of q-algebra representations: Tensor products of special unitary and oscillator algebras
Journal of Mathematical Physics, 1992
This paper begins a study of one-and two-variable function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebras considered are the quantum algebra U,(su2) and a q analog of the oscillator algebra (not a quantum algebra). In each case a simple one-variable model of the positive discrete series of finite-and infinitedimensional irreducible representations is used to compute the Clebsch-Gordan coefficients. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the "group operators" on these representation spaces are computed. It is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations. It is also demonstrated that general q-hypergeometric functions can occur as basis functions in two-variable models, in contrast with the very restricted parameter values for the q-hypergeometric functions arising as matrix elements in the theory of quantum groups.
Zeitschrift f�r Physik C Particles and Fields, 1995
On a nonstandard two-parametric quantum algebra and its connections with U p,q (gl(2)) and U p,q (gl(1|1)) Abstract. A quantum algebra U p,q (ζ, H, X ± ) associated with a nonstandard R-matrix with two deformation parameters(p, q) is studied and, in particular, its universal R-matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandard R-matrix is obtained through a coloured generalized boson realization of the universal R-matrix of the standard U p,q (gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal R-matrix of U p,q (gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandard R-matrices. Superization of U p,q (ζ, H, X ± ) leads to the super-Hopf algebra U p,q (gl(1|1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebra U p,q (sh(1)) and its univerprovides an infinite dimensional representation of the algebra (3.1) in the Fock space F z |m z = a m + |0 z | a − |0 z = 0, N|0 z = 0, m ∈ Z Z + , z ∈ C C}:
New Deformation of quantum oscillator algebra: Representation and some application
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, alpha\alphaalpha, and nu\nunu. The time-independent Schr\"{o}dinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
(q;l,λ)-deformed Heisenberg algebra: representations, special functions and quantization
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This paper addresses a new characterization of Sudarshan's diagonal representation of the density matrix elements ρ(z',z), derivedfrom (q;l,λ)-deformed boson coherent states.The induced ρ(z',z) self-reproducing property with the associated self-reproducing kernel K(z',z) is computed and analyzed. An explicit construction of novel classes of generalized continuous (q;l,λ)-Hermite polynomials is provided with the corresponding recursion relations and exact resolution of the moment problems giving their orthogonality weight functions. Besides, the Berezin-Klauder-Toeplitz quantization of classical phase space observables and relevant normal and anti-normal forms are investigated and discussed.