scholarly journals Representation Theory and Wigner-Racah Algebra of the SU(2) Group in a Noncanonical Basis

2005 ◽  
Vol 70 (6) ◽  
pp. 771-796 ◽  
Author(s):  
Maurice R. Kibler
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Deformation Parameter ◽  
Polar Decomposition ◽  
Classical Group ◽  
Quantization Scheme ◽  
Common Root ◽  
Root Of Unity ◽  
Recoupling Coefficients ◽  
Racah Algebra

The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the generators J- and J+ of the SU(2) group, with J+ = J-† = HUr where H is Hermitean and Ur unitary, and (ii) an alternative to the {J2,Jz} quantization scheme, viz., the {J2,Ur} quantization scheme. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The key ideas for developing the Wigner-Racah algebra of the SU(2) group in the {J2,Ur} scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the {J2,Ur} scheme are examined in great detail.

scholarly journals ANGULAR MOMENTUM AND MUTUALLY UNBIASED BASES

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1792-1801 ◽  
Author(s):  
MAURICE R. KIBLER
Keyword(s):  
Angular Momentum ◽  
Lie Algebra ◽  
Deformation Parameter ◽  
Casimir Operator ◽  
Mutually Unbiased Bases ◽  
Quantization Scheme ◽  
Root Of Unity ◽  
Commuting Operators ◽  
Deformed Oscillator ◽  
Complete Set

The Lie algebra of the group SU 2 is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J2, Ur} scheme, an alternative to the familiar {J2, Jz} quantization scheme corresponding to common eigenvectors of the Casimir operator J2 and the Cartan operator Jz. A connection is established between the eigenvectors of the complete set of commuting operators {J2, Ur} and mutually unbiased bases in spaces of constant angular momentum.

scholarly journals Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

2020 ◽  
Author(s):  
Daniel Oeh
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Polar Decomposition ◽  
Complex Hilbert Space ◽  
Analytic Extension ◽  
Involutive Automorphism ◽  
Finite Dimensional ◽  
Continuous Unitary Representation ◽  
Continuous Representations ◽  
Connected Lie Group

Abstract Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

scholarly journals A q-OSCILLATOR WITH "ACCIDENTAL" DEGENERACY OF ENERGY LEVELS

2007 ◽  
Vol 22 (13) ◽  
pp. 949-960 ◽  
Author(s):  
A. M. GAVRILIK ◽  
A. P. REBESH
Keyword(s):  
Lie Algebra ◽  
Commutation Relation ◽  
Deformation Parameter ◽  
Uncertainty Relation ◽  
Energy Levels ◽  
The State ◽  
The Real ◽  
Simple Lie Algebra ◽  
Accidental Degeneracy ◽  
Real Value

We study main features of the exotic case of q-deformed oscillators (so-called Tamm–Dancoff cutoff oscillator) and find some special properties: (i) degeneracy of the energy levels En1 = En1 + 1, n1 ≥ 1, at the real value[Formula: see text] of deformation parameter, as well as the occurrence of other degeneracies En1 = En1 + k, for k ≥ 2, at the corresponding values of q which depend on both n1 and k; (ii) the position and momentum operators X and Pcommute on the state|n1> if q is fixed as [Formula: see text], that implies unusual uncertainty relation; (iii) two commuting copies of the creation, annihilation, and number operators of this q-oscillator generate the corresponding q-deformation of the non-simple Lie algebra su(2) ⊕ u(1)whose nontrivial q-deformed commutation relation is: [J+, J-] = 2J0q2J3-1 where [Formula: see text] and [Formula: see text].

scholarly journals A combinatorial identity for the derivative of a theta series of a finite type root lattice

2003 ◽  
Vol 172 ◽  
pp. 1-30
Author(s):  
Satoshi Naito
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Finite Type ◽  
Cartan Subalgebra ◽  
Theta Series ◽  
Combinatorial Identity ◽  
Root Lattice ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra

AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine Lie algebra.

scholarly journals QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

2006 ◽  
Vol 15 (10) ◽  
pp. 1245-1277 ◽  
Author(s):  
STEPHEN F. SAWIN
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Roots Of Unity ◽  
Quantum Field ◽  
Root Of Unity ◽  
Basic Representation ◽  
Topological Quantum ◽  
Quantum Version ◽  
Simply Connected ◽  
Connected Lie Group

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

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

2001 ◽  
Vol 03 (04) ◽  
pp. 533-548 ◽  
Author(s):  
NAIHUAN JING ◽  
KAILASH C. MISRA ◽  
CARLA D. SAVAGE
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Mathematical Physics ◽  
Infinite Dimensional ◽  
Algebra Representation ◽  
The Sixties ◽  
Lie Algebra Representation ◽  
New Interpretation

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.

scholarly journals Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Thomas Bliem ◽  
Dido Salazar
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Ehrhart Polynomial ◽  
International Audience ◽  
Order Polytope ◽  
Finite Partially Ordered Set ◽  
The Relationship

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.

UNITARY BRAID REPRESENTATIONS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 985-1006
Author(s):  
HANS WENZL
Keyword(s):  
Quantum Groups ◽  
Braid Group ◽  
Deformation Parameter ◽  
Hecke Algebras ◽  
Tensor Products ◽  
Roots Of Unity ◽  
Von Neumann ◽  
Root Of Unity ◽  
Brauer Algebras ◽  
Original Motivation

The original motivation for studying unitarizable representations of the braid group was to construct interesting examples of subfactors of II1 von Neumann factors. This is possible for representations factoring through Hecke algebras or q-Brauer algebras only if the deformation parameter is a root of unity. Information gained from this work can be applied to studying tensor products of representations of quantum groups at roots of unity and invariants of 3-manifolds.

scholarly journals Unified Treatment of a Class of Spherically Symmetric Potentials: Quasi-Exact Solution

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
H. Panahi ◽  
M. Baradaran
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Representation Theory ◽  
Exact Solutions ◽  
Lie Algebra ◽  
Algebraic Approach ◽  
Spherically Symmetric ◽  
Exact Solvability ◽  
Enveloping Algebra ◽  
Basic Differential Equation

We investigate the Schrödinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra ofsl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory ofsl(2)Lie algebra.

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