Lie Group SU Research Articles

Characterizing SU(1,1) nonclassicality via variance

We quantify the nonclassicality of quantum states associated with the Lie group SU(1,1) by regarding states as observables and considering their variances in the SU(1,1) Perelomov coherent states. Combining the resolution of identity induced by the SU(1,1) Perelomov coherent states, we propose a quantifier for nonclassicality of a state based on the average uncertainty (variance) of the state (regarded as an observable) in the SU(1,1) Perelomov coherent states. This quantifier is easy to calculate and possesses several operational interpretations. We reveal its basic properties and illustrate it by several prototypical examples.

On the relation between the three Reidemeister moves and the three gauge groups

Quantum theory suggests that the three observed gauge groups U(1), SU(2) and SU(3) are related to the three Reidemeister moves: twists, pokes and slides. The background for the relation is provided. It is then shown that twists generate the group U(1), whereas pokes generate SU(2). Emphasis is placed on proving the relation between slides, the Gell-Mann matrices, and the Lie group SU(3). Consequences for unification are deduced.

Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor

We present a novel approach to generate Bessel–Gauss modes of arbitrary integer order and well-defined optical angular momentum in a gradient index medium of transverse parabolic profile. The propagation and coherence properties, as well as the quality factor, are studied using algebraic techniques that are widely used in quantum mechanics. It is found that imposing the well-defined optical angular momentum condition, the Lie group SU(1,1) comes to light as a characteristic symmetry of the Bessel–Gauss beams.

Mathematical Aspects of <i>SU</i> (2) and <i>SO</i>(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

Topologically conjugate classifications of the translation actions on compact connected Lie groups SU(2) × Tn

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups . We define the rotation vectors of the left actions induced by the elements in the maximal tori of , and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered.

Quantum gravity at the fifth root of unity

We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU(2) quantum group deformation symmetry, where the deformation parameter is postulated to be a complex fifth root of unity. By considering fermionic cycles through the foam, we couple this SU(2) quantum group with the same deformation of SU(3) so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. In this case, the amplitudes have a polytope interpretation. The generalization to higher-dimensional Lie groups SU(N), G2 and E8 is suggested.

3-folds CR-embedded in 5-dimensional real hyperquadrics

E. Cartan’s method of moving frames is applied to 3-dimensional manifolds M which are CR-embedded in 5-dimensional real hyperquadrics Q in order to classify M up to CR symmetries of Q given by the action of one of the Lie groups SU(3,1) or SU(2,2). In the latter case, the CR structure of M derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity is discussed. In both cases, we compute which homogeneous CR 3-folds appear in Q.

Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space

We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1, 1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1, 1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra . We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. We explicitly discuss how the representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the representation label sets a lower cut-off for the loop quantum gravity spins (=U(1) representation labels in LQC). Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the representation label and interpreted as the minimally resolved geometric scale.

LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory

We present LieART 2.0 which contains substantial extensions to 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. The basic procedure is unchanged—it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15. Program summaryProgram Title: LieART 2.0CPC Library link to program files:http://dx.doi.org/10.17632/8vm7j67bwt.1Licensing provisions: GNU Lesser General Public LicenseProgramming language: MathematicaExternal routines/libraries: Wolfram Mathematica 8–12Nature 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), but it has found use in a variety of other applications. Tensor product decomposition and subalgebra decomposition have been implemented for all classical Lie groups SU(N), SO(N) and Sp(2N) and all the exceptional groups E6, E7, E8, F4 and G2. This includes both regular and irregular (special) subgroup decomposition of all Lie groups up through rank 15, and many more.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]. We generate projection matrices by the Dynkin procedure, i.e., removing dots from the Dynkin or extended Dynkin diagram, for regular subalgebras, and we implement explicit projection matrices for special subalgebras.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 physicist. 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 the used cutoff is too low and the assignment of primes is incorrect. However, this only affects the display of the irrep. All computations involving this irrep are correct, since the internal unique representation of Dynkin labels is used.

K-theory of AF-algebras from braided C*-tensor categories

Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans and Gould (1994) explicitly described the [Formula: see text]-theory of certain unital AF-algebras [Formula: see text] as (quotients of) polynomial rings. In this paper, we show that in each case the multiplication in the polynomial ring (quotient) is induced by a ∗-homomorphism [Formula: see text] arising from a unitary braiding on a C*-tensor category and essentially defined by Erlijman and Wenzl (2007). We also present some new explicit calculations based on the work of Gepner, Fuchs and others. Specifically, we perform computations for the rank two compact Lie groups SU(3), Sp(4) and G2 that are analogous to the Evans–Gould computation for the rank one compact Lie group SU(2). The Verlinde rings are the fusion rings of Wess–Zumino–Witten models in conformal field theory or, equivalently, of certain related C*-tensor categories. Freed, Hopkins and Teleman (early 2000s) realized these rings via twisted equivariant [Formula: see text]-theory. Inspired by this, our long-term goal is to realize these rings in a simpler [Formula: see text]-theoretical manner, avoiding the technicalities of loop group analysis. As a step in this direction, we note that the Verlinde rings can be recovered as above in certain special cases.

On constant solutions of SU(2) Yang-Mills equations with arbitrary current in Euclidean space ℝn

In this paper, we present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space ℝn of arbitrary finite dimension n. Using the invariance of the Yang-Mills equations under the orthogonal transformations of coordinates and gauge invariance, we choose a specific system of coordinates and a specific gauge fixing for each constant current and obtain all constant solutions of the Yang-Mills equations in this system of coordinates with this gauge fixing, and then in the original system of coordinates with the original gauge fixing. We use the singular value decomposition method and the method of two-sheeted covering of orthogonal group by spin group to do this. We prove that the number (0, 1, or 2) of constant solutions of the Yang-Mills equations in terms of the strength of the Yang-Mills field depends on the singular values of the matrix of current. The explicit form of all solutions and the invariant F2 can always be written using singular values of this matrix. The relevance of the study is explained by the fact that the Yang-Mills equations describe electroweak interactions in the case of the Lie group SU(2). Non-constant solutions of the Yang-Mills equations can be considered in the form of series of perturbation theory. The results of this paper are new and can be used to solve some problems in particle physics, in particular, to describe physical vacuum and to fully understand a quantum gauge theory.

The algebraic structure of the SU(7) Lie group

In recent years, the Lie group SU(7) has been featured prominently in a number of grand unification proposals involving the Standard Model as a low energy effective theory. This note investigates the framework of the SU(7) group. The antisymmetric and symmetric structure constants for the Lie algebra su (7) have been explicitly calculated from the generators for the fundamental representation of SU(7), which are also cataloged.

Optimal control on SU(2) Lie group with stability analysis

Many configuration spaces of mechanical and non-mechanical problems in physical world involve matrix Lie groups. These Lie groups provide a mathematically rich formulation for studying a variety of control theory problems. While control systems have been specialized to different matrix Lie groups, study of optimal control of these systems by minimizing the input cost function has been a tempting research area. Here we take a left invariant driftless control system on the Lie group SU(2) and analyse controllability and optimal control of the system. Stability of the resulting dynamics from optimal control analysis is elaborated. Then numerical integration and some related properties are discussed via two unconventional integrators.

Quasi-Grammian solutions of the generalized Heisenberg magnet model

In this paper we use standard binary Darboux transformation to obtain the quasi-Grammian multi-soliton solutions of generalized Heisenberg magnet model in two dimensions. We also discuss the model based on the Lie group SU(n) and obtain explicit solutions of the model for the SU(2) case.

On conjugation orbits of semisimple pairs in rank one

Abstract We consider the Lie groups SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} and Sp ⁢ ( n , 1 ) {\mathrm{Sp}(n,1)} that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in Sp ⁢ ( n , 1 ) {\mathrm{Sp}(n,1)} and SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} up to conjugacy. This gives local parametrization of the representations ρ in Hom ⁢ ( F 2 , G ) / G {{\mathrm{Hom}}({\mathrm{F}}_{2},G)/G} such that both ρ ⁢ ( x ) {\rho(x)} and ρ ⁢ ( y ) {\rho(y)} are semisimple elements in G, where F 2 = 〈 x , y 〉 {{\mathrm{F}}_{2}=\langle x,y\rangle} , G = Sp ⁢ ( n , 1 ) {G=\mathrm{Sp}(n,1)} or SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} . We use the PSp ⁢ ( n , 1 ) {{\mathrm{PSp}}(n,1)} -configuration space M ⁢ ( n , i , m - i ) {{\mathrm{M}}(n,i,m-i)} of ordered m-tuples of distinct points in 𝐇 ℍ n ¯ {\overline{{\mathbf{H}}_{{\mathbb{H}}}^{n}}} , where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on M ⁢ ( n , i , m - i ) {{\mathrm{M}}(n,i,m-i)} .

Decomposition matrices for the square lattices of the Lie groups SU(2)times SU(2)

A method for the decomposition of data functions sampled on a finite fragment of rectangular lattice is described. The symmetry of a square lattice in a 2-dimensional real Euclidean space is either given by the semisimple Lie group SU(2)times SU(2) or equivalently by the Lie algebra A_1times A_1, or by the simple Lie group O(5) or its Lie algebra called C_2 or equivalently B_2. In this paper we consider the first of these possibilities which is applied to data which is given in 2 orthogonal directions—hence the method is a concatenation of two 1-dimensional cases. The asymmetry we underline here is a different density of discrete data points in the two orthogonal directions which cannot be studied with the simple Lie group symmetry.

Non-Abelian momentum polytopes for products of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{CP}^2 $\end{document}</tex-math></inline-formula>

This is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the point vortex systems familiar in 2-D. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 4-space. For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a small number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope, and numerous transition polytopes, all of which are classified here. The type of polytope depends on the weights of the symplectic form on each copy of projective space. In the second paper we use techniques of symplectic reduction to study the possible dynamics of interacting generalized point vortices. The results can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices with double eigenvalues.

SU(p,q) coherent states and a Gaussian de Finetti theorem

We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-mode Fock space. The (Gaussian) SU(p, q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n × n upper-left submatrix of an n × n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 = O(m).

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups SU(n) for n≥6, SO(n) for n≥7, Sp(n) for n≥3, E6,E7,E8, and F4 admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.

On the intersection pairing between cycles in SU(p,q)-flag domains and maximally real Schubert varieties

An SU(p,q)-flag domain is an open orbit of the real Lie group SU(p,q) acting on the complex flag manifold associated to its complexification SL(p+q,C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. Equivalently, the Borel group contains the AN-factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived.