Bruno Gruber

CLASSIFICATION OF SEMISIMPLE SUBALGEBRAS OF SIMPLE LIE ALGEBRAS.

An explicit classification of the semisimple complex Lie subalgebras of the simple complex Lie algebras is given for algebras up to rank 6. The notion of defining vector, introduced by Dynkin and valid for subalgebras of rank 1, has been extended to the notion of defining matrix, valid for any semisimple subalgebra. All defining matrices have been determined explicitly, which is equivalent to the determination of the embeddings of the generators of the Cartan subalgebra of a semisimple subalgebra in the Cartan subalgebra K of the simple algebra containing this subalgebra. Moreover, the embedding of the root system of the subalgebras in the dual space K* of an algebra is given for all subalgebras. For the S‐subalgebras of the simple algebras (up to rank 6), the embedding of the whole subalgebra in an algebra is given explicitly. In addition, the decomposition (branching) of the defining (fundamental) and adjoint representations of an algebra with respect to the restriction to its S‐subalgebras has been det...

Symmetries in science II

This book constitutes the proceedings of a symposium entitled Symmetries in Science II. Beginning with a general review of symmetries in physics, fifty-one papers take the reader through the veiled worlds of symmetries and dynamics, time symmetry and measure theory, symmetries and polarization, symmetries in heavy nuclei, and the proton-neutron interaction. Contributors explore symmetry between bosons and fermions and the fermion dynamical symmetry model, nuclear shell model, and collective nuclear structure physics. Other contributions probe squeezed states, squeezed states and quadratic Hamiltonians, symmetry-breaking in biological cells, and observed manifestations of invariance in condensed matter and biological systems. Papers range from a non-equilibrium theory for phase transitions and the expanding universe to the basic concepts of symmetry applications and its consequences.

Matrix elements for infinitesimal operators of the groups U(p+q) and U(p,q) in a U(p) ×U(q) basis. I

In this article explicit expressions are obtained for the action of the infinitesimal operators of the principal nonunitary series representations of the groups U(p,q) in a U(p) ×U(q) basis. It is moreover shown how the finite dimensional irreducible representations of the group U(p,q) and the group U(p+q) with respect to a U(p) ×U(q) basis are obtained from the principal nonunitary series representations of the group U(p,q).

Symmetries in Science IX

Classical vs Quantum Groups as Symmetries of Quantized Systems M. Arik, G. Unel. Algebraic Model of an Oblate Top R.Bijker, A. Leviatan. The Mass-Squared Operator and the Einstein-Hilbert Action for Rescaled Lorentz Metrics E. Binz, P. Oellers. Multichannel Dynamic Symmetry J. Cseh. Kazhdan-Lusztig Polynomials, Subsingular Vectors and Conditionally Invariant q-Deformed Equations V.K. Dobrev. On a Path to Nonlinear Quantum Mechanics H.-D. Doebner, J.-D. Hennig. Quantum Mechanical Problems with q-Deformations and over the p-Adic Number Fields I.H. Duru. A Symmetry Adapted Algebraic Approach to Molecular Spectroscopy A. Frank, et al. Dyson Boson-Fermion Realization of Lie (Super)Algebras D.V. Fursa, et al. Formal Languages for Quasicrystals J.G. Escudero. On Quadratic and Nonquadratic Forms: Applications to R2m -> R2m-n Nonbijective Transformations M. Kibler. Quantization of Systems with Constraints J.R. Klauder. Automorphisms and Discrete Fiber Bundles P. Kramer, et al. Algebraic Approach to Baryon Structure A. Leviatan, R. Bijker. Discrete Reflection Groups and Induced Representations of Poincare Group on the Lattice M. Lorente. 10 Additional Articles. Index.

Matrix elements for indecomposable representations of complex su(2)

Indecomposable representations of the simple complex Lie algebra A1 are investigated in this article from a general point of view. First a ‘‘master representation’’ is obtained which is defined on the space of the universal enveloping algebra Ω of A1. Then, from this master representation other indecomposable representations are derived which are induced on quotient spaces or subduced on invariant subspaces. Finally, it is shown that the familiar finite‐dimensional and infinite‐dimensional irreducible representations of su(2) and su(1,1) are closely related to certain of the indecomposable representations. Indecomposable representations of A1 [su(2), su(1,1)] have found increased applications in physical problems, including the unusual ‘‘finite multiplicity’’ indecomposable representations. Emphasis is placed in this article on an analysis of the more unfamiliar indecomposable representations. The matrix elements are obtained in explicit form for all representations which are discussed in this article. Th...

Properties of linear representations with a highest weight for the semisimple Lie algebras

A theory for the representations with a highest and/or lowest weight is given for the semisimple complex Lie algebras (and their real forms). These representations are either irreducible finite‐dimensional, irreducible infinite‐dimensional or reducible, but not completely reducible, infinite‐dimensional (called elementary representations), depending upon the property of the associated highest (or lowest) weight Λ. No restriction is made to those representations of the semisimple Lie algebras which can be integrated to form representations of the corresponding Lie group. The algebra A1 is chosen (Sec. III) as a simple and familiar example upon which, however, much of the proof for the results obtained for the theory of representations with a highest (and/or lowest) weight for the general case of a semisimple Lie algebra rests (Sec. IV). It is demonstrated that the irreducible representations D (Λ) with a highest (and/or lowest) weight Λ of the semisimple Lie algebras decompose with respect to any (regularl...

Boson operator realizations of su(2) and su(1,1) and unitarization

Boson operator realizations of su(2) and su(1,1) are obtained. Scalar products are introduced on ‘‘Fock spaces’’ (Verma modules) spanned by generators of the Heisenberg algebra H and by generators of su(2). These scalar products unitarize certain of the representations of H, or of su(1,1). It is shown that the Gel’fand–Dyson realization of su(1,1) implies a scalar product that unitarizes H, while the Primakoff–Holstein realizations imply a scalar product that unitarizes su(1,1). The relationship between the Gel’fand–Dyson boson operators a° and the Primakoff–Holstein boson operators b° is obtained making use of the two distinct scalar products. Generalized ‘‘vacuum states’’ are defined that are formed by polynomials in the creation–annihilation operator pairs a°a. A representation ρ of H and su(1,1) on the states a°m and an is discussed. For this representation a1=a‖0〉≠0, but rather (a°a)‖0〉 =0. The states of this representation space consist of boson states and boson‐hole states. All the familiar results...

Structure and matrix elements of the degenerate series representations of U(p+q) and U(p,q) in a U(p)×U(q) basis

The representations of the most degenerate series of the group U(p,q) which are induced by the representations of the maximal parabolic subgroup are considered in this article. By making use of the infinitesimal operators of these representations in the U(p)×U(q) basis the conditions are derived which are necessary and sufficient for irreducibility. For the reducible representations we describe their structure (composition series). We select from among the irreducible representations which are obtained in this article all representations of U(p,q) which admit unitarization. As a result we obtain the principal degenerate series, the supplimentary degenerate series, the discrete degenerate series, and the exceptional degenerate series of unitary representations of U(p,q). The U(p)×U(q) spectrum of the representations of U(p+q) with highest weights (λ1, 0,..., 0, λ2) is defined. We obtain the integral representation for the matrix elements of the degenerate representations of U(p,q) in the U(p)×U(q) basis. T...

Multiplicity free and finite multiplicity indecomposable representations of the algebra su(1,1)

A classification is given for the multiplicity free indecomposable representations of the simple Lie algebra su(1,1), which are unbounded on both sides. Formulas have been obtained for the matrix elements of the generators of su(1,1) for all these representations. Representations of su(1,1) are analyzed which have the property that all their weight subspaces are infinite dimensional. Subrepresentations and representations on quotient spaces of this infinite multiplicity representations are considered and their relationship to the multiplicity free indecomposable representations is determined (both, unbounded on both sides, and bounded on one side). Finite multiplicity indecomposable representations are obtained from the infinite multiplicity representation for special values of the Casimir operator. A decomposition of the infinite multiplicity representation into a direct sum of multiplicity free representations and finite multiplicity indecomposable respresentations is given in two different ways. Finall...

Indecomposable representations ofA2(su3,su2,1) and representations induced by them

SummaryIndecomposable representations are investigated for the case of the simple complex Lie algebraA2 (the complexification ofsu3 andsu2,1). The matrix elements are explicitly determined for the elementary representations, and the extremal vectors which characterize invariant subspaces are given in explicit form. Quotient spaces are used to derive other representations from the elementary representations, including the finite-dimensional irreducible representations, the infinite-dimensional irreducible representations which are bounded above, as well as new types of indecomposable representations. Again, the matrix elements for these representations are given in explicit form. In the appendix the same program is carried out for the simple complex lie algebraA1 (the complexification ofsu2 andsu1,1) as an example. The branching of the elementary representations, as well as of the representations derived from the elementary representations, is analyzed with respect to two subalgebras of the typeA1. Again, theA1 extremal vectors are obtained in explicit form.RiassuntoSi studiano le rappresentazioni non scomponibili per il caso della semplice algebra di Lie complessaA2 (la complessificazione disu3 esu2,1). Si determinano esplicitamente gli elementi matriciali per le rappresentazioni elementari e si danno in forma esplicita i vettori estremali che caratterizzano sottospazi invarianti. Si usano gli spazi quoziente per derivare altre rappresentazioni da quelle elementari, comprese le rappresentazioni irriducibili a dimensione finita, le rappresentazioni irriducibili a dimensione infinita limitate superiorimente, ed anche nuovi tipi di rappresentazioni non scomponibili. Si danno inoltre in forma esplicita gli elementi matriciali per queste rappresentazioni. In appendice si esegue lo stesso programma per la semplice algebra di Lie complessaA1 (la complessificazione disu2 esu1,1) come esempio. Si analizza la diramazione delle rappresentazioni elementari, ed anche delle rappresentazioni derivate da quelle elementari, rispetto a due subalgebre di tipoA1. Si ottengono inoltre i vettori estremali diA1 in forma esplicita.РезюмеИсследуются неприводимые представления для случая простой комплексной алгебры ЛиA2(su3,su2,1). Матричные элементы выражаются в явном виде для элементарных представлений. В явном виде приводятся экстремальные векторы, которые характеризуют инвариантные подпространства. Используются частные пространства для вывода других представлений из элементармых представлений, включая конечномерные неприводимые представления, бесконечномерные неприводимые представления, которые являются ограниченными, а также новые типы неприводимых представлений. Матричные элементы для этих представлений записываются в явном виде. В приложении, как пример, реализуется та же программа для простой комплексной алгебры ЛиA2(su2,su1,1). Ветвление элементарных представлений, а также представлений, выведенных из элементарных представлений, анализируется относительно двух субалгебр типаA1. Экстремальные векторыA1 получаются в явном виде.