206 research outputs found
The Jordanian deformation of su(2) and Clebsch-Gordan coefficients
Representation theory for the Jordanian quantum algebra U=U_h(sl(2)) is
developed using a nonlinear relation between its generators and those of sl(2).
Closed form expressions are given for the action of the generators of U on the
basis vectors of finite dimensional irreducible representations. In the tensor
product of two such representations, a new basis is constructed on which the
generators of U have a simple action. Using this basis, a general formula is
obtained for the Clebsch-Gordan coefficients of U. It is shown that the
Clebsch-Gordan matrix is essentially the product of a triangular matrix with an
su(2) Clebsch-Gordan matrix. Using this fact, some remarkable properties of
these Clebsch-Gordan coefficients are derived.Comment: 8 pages, LaTeX. Presented at the 6th International Colloquium Quantum
Groups and Integrable Systems, Prague, June 199
The Hamiltonian H=xp and classification of osp(1|2) representations
The quantization of the simple one-dimensional Hamiltonian H=xp is of
interest for its mathematical properties rather than for its physical
relevance. In fact, the Berry-Keating conjecture speculates that a proper
quantization of H=xp could yield a relation with the Riemann hypothesis.
Motivated by this, we study the so-called Wigner quantization of H=xp, which
relates the problem to representations of the Lie superalgebra osp(1|2). In
order to know how the relevant operators act in representation spaces of
osp(1|2), we study all unitary, irreducible star representations of this Lie
superalgebra. Such a classification has already been made by J.W.B. Hughes, but
we reexamine this classification using elementary arguments.Comment: Contribution for the Workshop Lie Theory and Its Applications in
Physics VIII (Varna, 2009
Finite dimensional representations of the quantum group using the exponential map from
Using the Fronsdal-Galindo formula for the exponential mapping from the
quantum algebra to the quantum group , we show
how the -dimensional representations of can be obtained
by `exponentiating' the well-known -dimensional representations of
for ; 1/2 corresponds to the
defining 2-dimensional -matrix. The earlier results on the
finite-dimensional representations of and (or )
are obtained when . Representations of
\C \backslash \R and are also
considered. The structure of the Clebsch-Gordan matrix for is
studied. The same Clebsch-Gordan coefficients are applicable in the reduction
of the direct product representations of the quantum group .Comment: 17 pages, LaTeX (latex twice), no figures. Changes consist of more
general formula (4.13) for T-matrices, explicit Clebsch-Gordan coefficients,
boson realization of group parameters, and typographical correction
Realizations of and and generating functions for orthogonal polynomials
Positive discrete series representations of the Lie algebra and the
quantum algebra are considered. The diagonalization of a
self-adjoint operator (the Hamiltonian) in these representations and in tensor
products of such representations is determined, and the generalized
eigenvectors are constructed in terms of orthogonal polynomials. Using simple
realizations of , , and their representations, these
generalized eigenvectors are shown to coincide with generating functions for
orthogonal polynomials. The relations valid in the tensor product
representations then give rise to new generating functions for orthogonal
polynomials, or to Poisson kernels. In particular, a group theoretical
derivation of the Poisson kernel for Meixner-Pollaczak and Al-Salam--Chihara
polynomials is obtained.Comment: 20 pages, LaTeX2e, to appear in J. Math. Phy
Quantum communication through a spin chain with interaction determined by a Jacobi matrix
We obtain the time-dependent correlation function describing the evolution of
a single spin excitation state in a linear spin chain with isotropic
nearest-neighbour XY coupling, where the Hamiltonian is related to the Jacobi
matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case
an arbitrary element of the correlation function is expressed in a simple
closed form. Its asymptotic limit corresponds to the Jacobi matrix of the
Charlier polynomial, and may be understood as a unitary evolution resulting
from a Heisenberg group element. Correlation functions for Hamiltonians
corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials
are also studied. For the Hahn polynomials we obtain the general correlation
function, some of its special cases, and the limit related to the Meixner
polynomials, where the su(1,1) algebra describes the underlying symmetry. For
the cases of dual Hahn and Racah polynomials the general expressions of the
correlation functions contain summations which are not of hypergeometric type.
Simplifications, however, occur in special cases
Convolutions for orthogonal polynomials from Lie and quantum algebra representations
The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials
as overlap coefficients in the positive discrete series representations of the
Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to
generalisations of the convolution identities for these polynomials. Using the
Racah coefficients convolution identities for continuous Hahn, Hahn and Jacobi
polynomials are obtained. From the quantised universal enveloping algebra for
su(1,1) convolution identities for the Al-Salam and Chihara polynomials and the
Askey-Wilson polynomials are derived by using the Clebsch-Gordan and Racah
coefficients. For the quantised universal enveloping algebra for su(2) q-Racah
polynomials are interpreted as Clebsch-Gordan coefficients, and the
linearisation coefficients for a two-parameter family of Askey-Wilson
polynomials are derived.Comment: AMS-TeX, 31 page
Solutions of the compatibility conditions for a Wigner quantum oscillator
We consider the compatibility conditions for a N-particle D-dimensional
Wigner quantum oscillator. These conditions can be rewritten as certain triple
relations involving anticommutators, so it is natural to look for solutions in
terms of Lie superalgebras. In the recent classification of ``generalized
quantum statistics'' for the basic classical Lie superalgebras
[math-ph/0504013], each such statistics is characterized by a set of creation
and annihilation operators plus a set of triple relations. In the present
letter, we investigate which cases of this classification also lead to
solutions of the compatibility conditions. Our analysis yields some known
solutions and several classes of new solutions.Comment: 9 page
A class of infinite-dimensional representations of the Lie superalgebra osp(2m+1|2n) and the parastatistics Fock space
An orthogonal basis of weight vectors for a class of infinite-dimensional
representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is
introduced. These representations are particular lowest weight representations
V(p), with a lowest weight of the form [-p/2,...,-p/2|p/2,...,p/2], p being a
positive integer. Explicit expressions for the transformation of the basis
under the action of algebra generators are found. Since the relations of
algebra generators correspond to the defining relations of m pairs of
parafermion operators and n pairs of paraboson operators with relative
parafermion relations, the parastatistics Fock space of order p is also
explicitly constructed. Furthermore, the representations V(p) are shown to have
interesting characters in terms of supersymmetric Schur functions, and a simple
character formula is also obtained.Comment: 15 page
A classification of generalized quantum statistics associated with basic classical Lie superalgebras
Generalized quantum statistics such as para-statistics is usually
characterized by certain triple relations. In the case of para-Fermi statistics
these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1);
in the case of para-Bose statistics they are associated with the Lie
superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition
of ``a generalized quantum statistics associated with a classical Lie algebra
G'' was given, and a complete classification was obtained. Here, we consider
the definition of ``a generalized quantum statistics associated with a basic
classical Lie superalgebra G''. Just as in the Lie algebra case, this
definition is closely related to a certain Z-grading of G. We give in this
paper a complete classification of all generalized quantum statistics
associated with the basic classical Lie superalgebras A(m|n), B(m|n), C(n) and
D(m|n)
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