Representation Theory Of Lie Algebras Research Articles

Moment Systems Derived from Relativistic Kinetic Equations

In this paper, we are interested in the derivation of macroscopic equations from kinetic ones using a moment method in a relativistic framework. More precisely, we establish the general form of moments that are compatible with the Lorentz invariance and derive a hierarchy of relativistic moment systems from a Boltzmann kinetic equation. The proof is based on the representation theory of Lie algebras. We then extend this derivation to the classical case and general families of moments that obey the Galilean invariance are also constructed. It is remarkable that the set of formal classical limits of the so-obtained relativistic moment systems is not identical to the set of classical moments quoted in Ref. 21 and one could use a new physically relevant criterion to derive suitable moment systems in the classical case. Finally, the ultra-relativistic limit is considered.

Invariant differential operators and characters of the AdS4algebra

The aim of this paper is to apply systematically to AdS_4 some modern tools in the representation theory of Lie algebras which are easily generalised to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models. Here we introduce the necessary representations of the AdS_4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS_4 Verma modules. These are used to obtain the AdS_4 invariant differential operators. Using this we display a new structure - a diagram involving four partially equivalent reducible representations one of which contains all finite-dimensional irreps of the AdS_4 algebra. We study in more detail the cases involving UIRs, in particular, the Di and the Rac singletons, and the massless UIRs. In the massless case we discover the structure of sets of 2s_0-1 conserved currents for each spin s_0 UIR, s_0=1,3/2,... All massless cases are contained in a one-parameter subfamily of the quartet diagrams mentioned above, the parameter being the spin s_0. Further we give the classification of the so(5,C) irreps presented in a diagramatic way which makes easy the derivation of all character formulae. The paper concludes with a speculation on the possible applications of the character formulae to integrable models.

The Eigenfunction Expansion Method in Multi-factor Quadratic Term Structure Models

We propose the eigenfunction expansion method for pricing options in linear-quadratic terms structure models. The eigenvalues, eigenfunctions and adjoint functions are calculated using elements of the representation theory of Lie algebras not only in the self-adjoint case but in non-selfadjoint case as well; the eigenfunctions and adjoint functions are expressed in terms of the Hermite polynomials. We demonstrate that the method is efficient for pricing caps, floors and swaptions, if time to maturity is 1 year or more. We also consider the subordination of the same class of models, and show that in the framework of the eigenfunction expansion approach, the subordinated models are (almost) as simple as pure Gaussian models. We study the dependence of option prices on the type of non-Gaussian innovations, and suggest parameters' fitting procedures based on the properties of the asymptotic expansions.

Lie algebra representations, nilpotent matrices, and the C–numerical range

In this paper we survey some applications of the representation theory of Lie algebras to Linear Algebra. This includes the derivation of the Jordan form for tensor products of invertible matrices, the study of normal form problems for nilpotent matrices, as well as the derivation of explicit formulas for the C-numerical radius of certain nilpotent block-shift matrices, arising in quantum mechanics and nuclear magnetic resonance (NMR-) spectroscopy. In this latter case, a conjecture concerning an explicit formula for the C-numerical radius is stated. We show the existence of unitary transformations that realize the prospective maxima. Our approach depends on the Clebsch-Gordan decomposition for unitary representations of su 2 ( C ) .

Eigenfunction Expansion Method in Multi-factor Models

We show that three classes of multi-factor gaussian mean-reverting models: for the dynamics of the (log-)price of a stock, ATSM of the Ornstein-Uhlenbeck type, and QTSM are equivalent, when contingent claims with deterministic life-spans are considered. We provide the reduction of these models to two basic types of QTSMs, which are explicitly solved by the eigenfunction expansion technique. The reduction uses solutions to continuous algebraic Riccati equations and Lyapunov equations. The eigenvalues, eigenfunctions and adjoint functions are calculated by using elements of the representation theory of Lie algebras; the eigenfunctions and adjoint functions are expressed in terms of the Hermite polynomials. We also consider the same classes of models in random time, and show that in the framework of the eigenfunction expansion approach, the models in random time are (almost) as simple as pure gaussian models. We suggest parameters' fitting procedures based on the properties of the asymptotic expansions.

Solution to the Master Equation of a Free Damped HarmonicOscillator with Linear Driving

We use the Lie algebra representation theory for superoperators to solvethe master equation for a harmonic oscillator with a linear driving termin a squeezed thermal reservoir. By using the quantum displacementtransformation and squeeze transformation, we show that the masterequation has an su(1,1) Lie algebra structure, with which we obtainthe explicit solution to the master equation. A simple but typicalexample is given to illustrate our method.

Weight sum formulae in Lie algebra representations

Many formulae in the representation theory of Lie algebras involve sums of weights, the most celebrated being that of H. Weyl [Math. Z. 24 (1926) 377–395]. In this note we will derive some elementary, but very useful, formulae for summing subsets of weights of a Lie algebra representation. We also give a selection of applications of these formulae.

Quantum many-body problems and perturbation theory

We show that the existence of algebraic forms of exactly-solvable $A-B-C-D$ and $G_2, F_4$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theory based on representation theory of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.

ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

Multi-Sided Braid Type Subfactors

AbstractWe generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II1 factor R in [E1]—which arise by considering unitary braid representations with certain properties—to multisided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types A, B, C, D. We also compute the (first) relative commutants.

New subfactors from braid group representations

This paper is about the construction of new examples of pairs of subfactors of the hyperfinite II 1 _{1} factor, and the computation of their indices and relative commutants. The construction is done in general by considering unitary braid representations with certain properties that are satisfied in natural examples. We compute the indices explicitly for the particular cases in which the braid representations are obtained in connection with representation theory of Lie algebras of types A,B,C,D.

Forty-five years of many-body theory: What is the relation of field theory and representations of Lie algebras?

Abstract In celebration of David Pines's (70 + n)th birthday, this essay asks the question whether there is a truly fundamental relationship between the Hilbert spaces of a quantum theory and the representation theory of Lie algebras. In trying to sharpen this question we briefly review representation theory and then conclude with a well known example where the Hilbert space of a simple field theory is an irreducible representation of a Lie algebra.

-Categories and -Functors in the Representation Theory of Lie Algebras

-Categories and -Functors in the Representation Theory of Lie Algebras

𝔉-categories and 𝔉-functors in the representation theory of Lie algebras

The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. These include of course Horn, Ext, and Tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple Lie algebra. In this article we let a \mathfrak {a} be a Lie subalgebra of a Lie algebra g \mathfrak {g} and Γ \Gamma be a functor on some category of a \mathfrak {a} -modules. We then consider the following general question: For a g \mathfrak {g} -module E what hypotheses on Γ \Gamma and E are sufficient to insure that Γ ( E ) \Gamma (E) admits a canonical structure as a g \mathfrak {g} -module? The article offers an answer through the introduction of the notion of F \mathfrak {F} -categories and F \mathfrak {F} -functors. The last section of the article treats various examples of this theory.

Analogues of Riemann tensors for the odd metric on supermanifolds

Structure functions constitute the complete set of obstructions to integrability of a G-structure on a manifold. For a Riemannian manifold the structure function is the Riemann tensor. In this work, we compute structure functions for the odd analogue of the metric on supermanifolds and for several related structures. Structure functions take values in Spencer cohomology groups, which we describe by means of the representation theory of Lie algebras and Lie superalgebras.

Sharply transitive linear algebraic groups

We study Zariski-closed linear groupsG ≤ GL n (k) over fieldsk of characteristic 0 which act sharply transitively on the non-zero vectors ofk n . For square-freen, orn⩽15, or ifk has cohomological dimension ⩽1 we obtain a complete classification (i.e. a reduction to questions about associative division algebras). The main tools are representation theory of Lie algebras over algebraically closed and non-closed fields, and results about simple associative algebras in order to control the interplay between linear Lie algebras and the associative algebras generated by them. The relation to nearfields and left-symmetric division algebras is also discussed.

Auslander Algebras as Quasi-Hereditary Algebras

The notion of a quasi-hereditary algebra has been introduced by E. Cline, B. Parshall and L. Scott [7,2,5] in order to describe the so-called highest weight categories arising in the representation theory of Lie algebras and algebraic groups. Quasi-hereditary algebras are defined by the existence of a suitable chain of ideals, and the finite dimensional hereditary algebras are typical examples. In [3], also finite dimensional algebras of global dimension 2 are shown to be quasi-hereditary. Thus, the Auslander algebras are quasi-hereditary. Recall that the Auslander algebras A can be constructed in the following way. Let R be a representation-finite finite dimensional algebra; then A is the endomorphism algebra End(Affl), where M is a finite dimensional /{-module such that every indecomposable /{-module is isomorphic to a direct summand of M. We are going to introduce the notion of a splitting filtration on the class of all indecomposable /{-modules and show that in this way we obtain a heredity chain of ideals of A (see the definition below). Usually, there exist many splitting filtrations for a given R. Examples of splitting filtrations can be obtained from the Rojter measure, used by A. V. Rojter in his proof of the first Brauer-Thrall conjecture [6], or from the preprojective and preinjective partitions, introduced by M. Auslander and S. Smalo in [1]. Instead of dealing with finite dimensional algebras, we shall consider, more generally, semiprimary rings. Recall that an associative ring A with 1 is called semiprimary provided that its Jacobson radical TV is nilpotent and A/N is semisimple artinian. We say that an ideal J of A is a heredity ideal of A if P = J, JNJ = 0 and /, considered as a right ideal, is a projective ^-module. Following [2], a semiprimary ring A is said to be quasi-hereditary provided that there exists a chain

Modular Representation Theory of Lie Algebras

On etudie les representations non necessairement restreintes de l'algebre universelle enveloppante U(#7B-G) pour une algebre de Lie restreinte #7B-G

Normal form and representation theory

Representation theory of Lie algebras is called upon to develop a procedure for normalizing a dynamical system with two degrees of freedom in the neighborhood of an equilibrium when the Hamiltonian H(x, y, X, Y) in the coordinates (x, y) and their conjugate momenta (X, Y) is of the type H=(X2+Y2)/2+V(x, y, X, Y), the potential energy V being a sum of homogeneous polynomials in the phase variables of degree strictly greater than 2. The fact that the resulting potential V′ is a polynomial in the new coordinates (x′, y′) and the angular momentum G′=x′Y′−y′X′ implies that the normalization is a rotation in the configuration space from a fixed frame to an ideal frame. The technique is intended for normalizing an Hamiltonian in equilibrium at the origin when the Lie derivative associated with the quadratic part is not semisimple, e.g., the planar restricted problem of three bodies at the equilateral equilibrium L4 when the basic frequencies are equal (Routh’s singular case).

Lie theory and difference equations. I

A factorization method is constructed for sequences of second-order linear difference equations in analogy with the factorization method for differential equations. Six factorization types are established and recursion relations are obtained for various classes of special functions, among which are the hypergeometric functions and their limits, and the classical polynomials of a discrete variable: Tchebycheff, Krawtchouk, Charlier, Meixner, and Hahn. It is shown that the factorization method is a disguised form of Lie algebra representation theory.