Representation of Lie Groups and Special Functions

Vilenkin, N. Ja.; Klimyk, A. U.

doi:10.1007/978-94-011-3538-2

Cited by 527 publications

(619 citation statements)

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“…13 Connections between group representations and special functions are explored in refs. [ 12,19 ]. Representations of the rotation group play a central role in quantum mechanics.…”

Section: Thenmentioning

confidence: 99%

“…[ 71 ] is based on the concatenation of a finite number of noisy motions. In the limiting case of a time-parameterized path of noisy motions, the covariance propagation formula can be written as (19) where Ad g (·) is the adjoint. Technically, the mean path, g m (t) is not the same as the noiseless path that can be obtained by solving the deterministic model.…”

Section: Shifted Gaussian Solution For Fokker-planck Equations With Smentioning

confidence: 99%

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Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Park¹,

Liu²,

et al. 2008

Robotica

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SUMMARYA nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker-Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space, SO(3), and the Euclidean motion group of the plane, SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

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“…13 Connections between group representations and special functions are explored in refs. [ 12,19 ]. Representations of the rotation group play a central role in quantum mechanics.…”

Section: Thenmentioning

confidence: 99%

Section: Shifted Gaussian Solution For Fokker-planck Equations With Smentioning

confidence: 99%

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Park¹,

Liu²,

et al. 2008

Robotica

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“…[26], have found an interpretation on compact quantum groups analogous to the interpretation of orthogonal polynomials of hypergeometric type from the Askey scheme on compact Lie groups and related structures, see e.g. [46,47].…”

Section: Introductionmentioning

confidence: 99%

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

2016

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Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (SU(2) × SU(2), diag) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey-Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials. B Pablo Román

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“…For 3j-and 6j-coefficients of su (2) there exist expressions in terms of hypergeometric series [5,6,7], explaining the close relation with orthogonal polynomials such as Hahn and Racah polynomials [7]. For example, the 6j-coefficient of su (2) is expressed in terms of a terminating balanced 4 F 3 series of unit argument.…”

Section: Introductionmentioning

confidence: 99%

“…The parameters of the 4 F 3 (1) series are written in terms of the six representation labels (angular momenta) of the 6j-coefficient. By the nature of these representation labels (integer or half-integer positive numbers), the parameters of the 4 F 3 (1) series are integers [6,7]. When identifying the 6j-coefficient with a Racah polynomial R m (λ(x); α, β, γ, δ), it is not easy to decide which parameters correspond to the degree m, which to the variable x, and which to the parameters α, β, γ, δ of the polynomial.…”

Section: Introductionmentioning

confidence: 99%

3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables

Lievens

Jeugt

2002

Journal of Mathematical Physics

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In the tensor product of n+1 positive discrete series representations of su(1, 1), a coupled basis vector can be described by a certain binary coupling tree. To every such binary coupling tree, polynomials R (k)l (x) and R (k) l (x) are associated. These polynomials are n-variable Jacobi and continuous Hahn polynomials, and are orthogonal with respect to a weight function. The connection coefficients expressing such a polynomial associated with a given binary coupling tree in terms of those polynomials associated with another binary coupling tree are proportional to 3nj-coefficients of su(1, 1).

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Representation of Lie Groups and Special Functions

Cited by 527 publications

References 0 publications

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables

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