Lie Group SU Research Articles

Adjoint representations for SU(2), su(2) and sl(2)

This work, presents four kinds of adjoint representations for the special unitary matrix Lie group SU(2) and the special unitary, special linear matrix Lie algebras su(2) and sl(2). In the first two we assume the vector spaces as the matrix Lie algebras su(2) and sl(2), later cases obtained by exploiting the action of su(2) and sl(2) on themselves. Also, we compute their direct sums. The results have been displayed as Tables in a nice form.

Astheno–Kähler and Balanced Structures on Fibrations

AbstractWe study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, strong Kähler with torsion (SKT), and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then we provide a construction of astheno-Kähler structures on torus bundles over Kähler manifolds leading to new examples. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metric, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups SU(3) and G2 admit SKT and astheno-Kähler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space M with invariant volume admits a balanced metric, then its first Chern class c1(M) does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.

Formula omitted] and [formula omitted] approaches to the radial oscillator: Generalized coherent states and squeezing of variances

It is shown that each one of the Lie algebras su(1,1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU(1,1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU(2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding generalized coherent states are constructed and the conditions to squeeze the involved quadratures are analyzed.

Sub-Riemannian distance in the Lie groups SU(2) and SO(3)

We calculate distances between arbitrary elements of the Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics ρ and d. In computing distances for the second metric, we substantially use the fact that the canonical two-sheeted covering epimorphism Ω of SU(2) onto SO(3) is a submetry and a local isometry in the metrics ρ and d. Despite the fact that the proof uses previously known formulas for geodesics starting at the unity, F. Klein’s formula for Ω, trigonometric functions, and the conventional differential calculus of functions of one real variable, we focus attention on a careful application of these simple tools in order to avoid the mistakes made in previously published mathematical works in this area.

Orbifold biquotients of [formula omitted

One of the main methods of constructing new spaces with positive or almost positive curvature is the study of biquotients first studied in detail by Eschenburg. In this paper we classify orbifold biquotients of the Lie Group SU(3), and construct a new example of a 5-dimensional orbifold with almost positive curvature. Furthermore, we extend the work of Florit and Ziller on the geometric properties of the orbifolds SU(3)//T2.

On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

Abstract This is a survey article about the study of the links of some complex hypersurface singularities in ℂ3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

On the construction of integrable surfaces on Lie groups

The problem of the immersion of a two-dimensional surface into a three-dimensional Euclidean space can be formulated in terms of the immersion of surfaces in Lie groups and Lie algebras. A general formalism for this problem is developed, as well as an equivalent Mauer–Cartan system of differential forms. The particular case of the Lie group SU(2) is examined, and it is shown to be useful for studying integrable surfaces. Some examples of such surfaces and their equations are presented at the end, in particular, the cases of constant mean curvature and of zero Gaussian curvature.

LieART—A Mathematica application for Lie algebras and representation theory

We present the Mathematica application “LieART” (LieAlgebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. LieART can handle all classical and exceptional Lie algebras. It computes root systems of Lie algebras, weight systems and several other properties of irreducible representations. LieART’s user interface has been created with a strong focus on usability and thus allows the input of irreducible representations via their dimensional name, while the output is in the textbook style used in most particle-physics publications. The unique Dynkin labels of irreducible representations are used internally and can also be used for input and output. LieART exploits the Weyl reflection group for most of the calculations, resulting in fast computations and a low memory consumption. Extensive tables of properties, tensor products and branching rules of irreducible representations are included as online supplementary material (see Appendix A). Program summaryProgram title: LieARTCatalogue identifier: AEVL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU Lesser General Public LicenseNo. of lines in distributed program, including test data, etc.: 183178No. of bytes in distributed program, including test data, etc.: 411658Distribution format: tar.gzProgramming language: Mathematica.Computer: x86, x86_64, PowerPC.Operating system: cross-platform.RAM:≥ 1 GB recommended. Memory usage depends strongly on the Lie algebra’s rank and type, as well as the dimensionality of the representations in the computation.Classification: 4.2, 11.1.External routines: Wolfram Mathematica 8-10Nature of problem: The use of Lie algebras and their representations is widespread in physics, especially in particle physics. The description of nature in terms of gauge theories requires the assignment of fields to representations of compact Lie groups and their Lie algebras. Mass and interaction terms in the Lagrangian give rise to the need for computing tensor products of representations of Lie algebras. The mechanism of spontaneous symmetry breaking leads to the application of subalgebra decomposition. This computer code was designed for the purpose of Grand Unified Theory (GUT) Model building, where compact Lie groups beyond the U(1), SU(2) and SU(3) of the Standard Model of particle physics are needed. Tensor product decomposition and subalgebra decomposition have been implemented for all classical Lie groups SU(N), SO(N) and Sp(2N) and the exceptionals E6, E7, E8, F4 and G2.Solution method: LieART generates the weight system of an irreducible representation (irrep) of a Lie algebra by exploiting the Weyl reflection groups, which is inherent in all simple Lie algebras. Tensor products are computed by the application of Klimyk’s formula, except for SU(N)’s, where the Young-tableaux algorithm is used. Subalgebra decomposition of SU(N)’s are performed by projection matrices, which are generated from an algorithm to determine maximal subalgebras as originally developed by Dynkin [1, 2].Restrictions: Internally irreps are represented by their unique Dynkin label. LieART’s default behavior in TraditionalForm is to print the dimensional name, which is the labeling preferred by physicists. Most Lie algebras can have more than one irrep of the same dimension and different irreps with the same dimension are usually distinguished by one or more primes (e.g. 175 and 175′ of A4). To determine the need for one or more primes of an irrep a brute-force loop over other irreps must be performed to search for irreps with the same dimensionality. Since Lie algebras have an infinite number of irreps, this loop must be cut off, which is done by limiting the maximum Dynkin digit in the loop. In rare cases for irreps of high dimensionality in high-rank algebras, if the cutoff used is too low, then the assignment of primes will be incorrect, but the problem can be avoided by raising the cutoff. However, in either case, this can only affect the display of the irrep because all computations involving this irrep are correct, since the internal unique representation of Dynkin labels is used.Running time: From less than a second to hours depending on the Lie algebra’s rank and type and/or the dimensionality of the representations in the computation.

Minimum time optimal synthesis for two level quantum systems

For the time optimal problem of an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU(2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge--Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.

ON THE DRESSING METHOD FOR THE GENERALIZED COUPLED DISPERSIONLESS INTEGRABLE SYSTEM

The dressing method of Zakharov and Shabat [Funct. Anal. Appl.8, 226 (1974) and ibid.13, 166 (1980)] has been employed to the generalized coupled dispersionless integrable system in two dimensions. The dressed solutions to the Lax pair and to the nonlinear matrix equation have been obtained in terms of Hermitian projectors. The dressing method has been related with the quasi-determinant solutions obtained by using the standard matrix Darboux transformation. The iteration of dressing procedure has been shown to give N-soliton solutions of the system. At the end, the explicit soliton solution has been obtained for the system based on Lie group SU(2).

Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control

We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.

Spinors group field theory and Voros star product: First contact

In the context of non-commutative geometries, we develop a group Fourier transform for the Lie group SU(2). Our method is based on the Schwinger representation of the Lie algebra su(2) in terms of spinors. It allows us to prove that the non-commutative R^3 space dual to the SU(2) group is in fact of the Moyal-type and endowed with the Voros star-product when expressed in the spinor variables. Finally, from the perspective of quantum gravity, we discuss the application of these new tools to group field theories for spinfoam models and their interpretation as non-commutative field theories with quantum-deformed symmetries.

Homogeneous Einstein metrics on [formula omitted

It is known that every compact simple Lie group admits a bi-invariant homogeneous Einstein metric. In this paper we use two ansatz to probe the existence of additional inequivalent Einstein metrics on the Lie group SU(n) for arbitrary n. We provide an explicit construction of (2k+1) inequivalent Einstein metrics on SU(2k) and 2k inequivalent Einstein metrics on SU(2k+1).

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D’Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E 6, E 7, and E 8.

On a Morita equivalence between the duals of quantum SU(2) and quantum [formula omitted

Let SUq(2) and E˜q(2) be Woronowicz's q-deformations of respectively the compact Lie group SU(2) and the non-trivial double cover of the Lie group E(2) of Euclidean transformations of the plane. We prove that, in some sense, their duals are ‘Morita equivalent locally compact quantum groups’. In more concrete terms, we prove that the von Neumann algebraic quantum groups L∞(SUq(2)) and L∞(E˜q(2)) are unitary cocycle deformations of each other.

Affine su(2) fusion rules from gerbe 2-isomorphisms

We give a geometric description of the fusion rules of the affine Lie algebra s u ̂ ( 2 ) k at a positive integer level k in terms of the k -th power of the basic gerbe over the Lie group SU ( 2 ) . The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of s u ̂ ( 2 ) k via a 1-isomorphism. The fusion-rule coefficients are related to the existence of a 2-isomorphism between pullbacks of these 1-isomorphisms to a submanifold of SU ( 2 ) × SU ( 2 ) determined by the corresponding three conjugacy classes. This construction is motivated by its application in the description of junctions of maximally symmetric defect lines in the Wess–Zumino–Witten model.

GRADED LIE ALGEBRA AND THE SU(3)L⊗U(1)N GAUGE MODEL

A classical gauge model based on the Lie group SU (3)L⊗ U (1)N with exotic quarks is reformulated within the formalism of nonassociative geometry associated with an L cycle. The N charges of the fermionic particles and the related parameter constraints are algebraic consequences and are uniquely determined. Moreover, the number of scalar particles is dictated by the nonassociativity of the geometry. As a byproduct of this formalism, the Weinberg angle θw, scalar, charged and neutral gauge boson masses, as well as the mixing angles, are derived. Furthermore, various expressions for the vector and axial couplings of the quarks and leptons with the neutral gauge bosons and lower bounds of the very heavy gauge bosons are obtained.

On the Construction of Quantum Homogeneous Spaces from ∗ -Galois Objects

In this note we construct bi- ∗ -Galois objects linking the quantized universal enveloping algebras associated to the Lie groups SU(2), E(2) and SU(1, 1), where E(2) denotes the Lie group of Euclidian transformations of the plane, and we show how one can create (formal) quantum homogeneous spaces for these quantum groups by integrating the associated Miyashita-Ulbrich action on certain subquotient ∗ -algebras.

DARBOUX TRANSFORMATION AND MULTI-SOLITON SOLUTIONS OF PRINCIPAL CHIRAL AND WZW MODELS

In this paper we present Darboux transformation for the principal chiral and WZW models in two dimensions and construct multi-soliton solutions in terms of quasideterminants. We also establish the Darboux transformation on the holomorphic conserved currents of the WZW model and expressed them in terms of the quasideterminant. We discuss the model based on the Lie group SU (n) and obtain explicit soliton solutions for the SU(2) model.