Classification of finite-dimensional modules over semisimple Lie algebras

Sophus Lie (1842-1899) known as the founder of the theory of transformation groups, originally aimed to study solutions of differential equations via their symmetries. Over the decades this theory has evolved into the theory of Lie groups. These Lie groups are of an analytic and geometric nature, but Sophus Lie's principal discovery was that...

Generalized positive energy representations of groups of jets

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra ř. Consider the Fréchet–Lie group G:= J0°(V; K) of i-jets at 0 e V of smooth maps V ! K, with Lie algebra g = 70° (V ; ř). Let P be a Lie group and write p:= Lie(P). Let a be a smooth P-action on G. We study smooth projective unitary...

Cohomological local-to-global principles and integration in finite- and infinite-dimensional Lie theory

The subject of this thesis is twofold: The first part is the study of local-to-global principles for the continuous Lie algebra (co-)homology of certain infinite-dimensional Lie algebras of geometric origin, specifically, Gelfand-Fuks cohomology and continuous cohomology of gauge algebras. It includes both an exposition to classical results of...

Multivariate generalisations of classical hypergeometric polynomials from Lie theory

In this thesis, we will be studying Lie groups and their connection to certain orthogonal polynomials. We will look into the classical Krawtchouk, Meixner and Laguerre polynomials, and the multivariate Krawtchouk and Meixner polynomials as defined by Iliev. Using representations of the Lie groups SU(2) and SU(1,1), it will be shown that the...

Representation theory of SU(2) and SU(3) with applications to spin and quark models

In this thesis we provide an elementary introduction in finite dimensional representation theory of the Lie groups SU(2) and SU(3) for undergraduate students in physics and mathematics. We will also give two application of representation theory of these two groups in physics: the spin and quark models. We begin with first discussing...

The L1-Algebra Of A Symplectic Manifold

We construct an L1-algebra on the truncated canonical homology complex of a symplectic manifold, which naturally projects to the universal central extension of the Lie algebra of Hamiltonian vector fields.

Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let gˆ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type BCˆn=A2n(2)). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with gˆ. In type Aˆn−1=An−1(1) the formula in question reproduces an affine Pieri rule for cylindric Hall...

Trajectory control and dynamic modeling of a tailless flapping-wing robot

The DelFly Nimble is a type of tailless flapping-wing micro air vehicles (FWMAVs) that has received an increasing amount of attention. FWMAVs show efficient and agile flight possibilities at small scale. The aerodynamics and dynamics of these flapping vehicles are challenging and not fully understood. In this work, a strategy to implement a 3D...

Orthogonal Dualities of Markov Processes and Unitary Symmetries

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related...

Properties of the Racahpolynomials with regard to the Lie algebrarepresentation of sl(2;C)

The Racah polynomial Rn(λ(x)) is a polynomial of degree n and is variable in λ(x). In this thesis two properties of this polynomial will be studied. One is the orthogonal property of the Racah polynomial. And the other is that the Racah polynomial can also be described as a polynomial of degree x and variable over λ(n). The Racah polynomials...

Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the underlying spherical symmetry

We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly,<br/>why the phase...

Orthogonal Stochastic Duality Functions from Lie Algebra Representations

We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between ∗-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the...

Lie algebra su(2) - Het energiespectrum van een waterstofatoom berekenen

Deze tekst geeft een introductie over de Lie algebra su(2). Met deze introductie, en met behulp van representaties en tensorproducten wordt uiteindelijk het energiespectrum van een waterstofatoom berekend.

Angular momentum dynamics and the intrinsic drift of monopolar vortices on a rotating sphere

On the basis of the angular momentum equation for a fluid shell on a rotating planet, we analyze the intrinsic drift of a monopolar vortex in the shell. Central is the development of a general angular momentum equation for Eulerian fluid mechanics based on coordinate-free, general tensorial representations of the underlying fluid dynamics on the...

Tensor product representations and special functions