Representations Of Compact Lie Groups Research Articles

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.

Representations of compact Lie groups of low cohomogeneity

We survey different tools to classify representations of compact Lie groups according to their cohomogeneity and apply these methods to the case of irreducible representations of cohomogeneity 6, 7 and 8.

Factorial Unitary Representations of Compact Lie Groups, Wigner Sets and Locally Invariant Quantum Fields

The fibre bundle construct defined in our previous work continues to be the context for this paper; quantum fields composed of fibre algebras become liftings of; or sections through; a fibre bundle with base space a subset of curved space-time. We consider a compact Lie group such as SU(n) acting as a local gauge group of automorphisms of each fibre algebra A(x). Compact Lie groups, represented as gauge groups acting locally on quantum fields, are key elements in electroweak and strong force unification. In our recent joint work we have focused on the translational subgroup of the Poincare group as the generator of local diffeomorphism invariant quantum states. Here we extend those algebraic non-perturbative approaches to address the other half of unification by considering the existence of quantum states of the fibre algebra A(x) invariant to the action of compact non-abelian Lie groups. Wigner sets are complementary to little groups and we prove they have the finite intersection property. Exploiting this then allows us to show that invariant states are common in the sense that the weakly closed convex hull of every normal (density matrix) state contains such an invariant state. From these results and our related research emerges the existence of a locally invariant density matrix quantum state of the field.

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.

An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds

We prove an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.

On orbit spaces of representations of compact Lie groups

Abstract. We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural results and apply them to study irreducible representations of small copolarity. As an important tool, we classify all irreducible representations of connected groups with cohomogeneity four or five.

Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces

We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schrodinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group

We consider asymptotic behavior of the dimension of the invariant subspace in a tensor product of several irreducible representations of a compact Lie group G . It is equivalent to studying the symplectic volume of the symplectic quotient for a direct product of several coadjoint orbits of G . We obtain two formulas for the asymptotic dimension. The first formula takes the form of a finite sum over tuples of elements in the Weyl group of G . Each term is given as a multiple integral of a certain polynomial function. The second formula is expressed as an infinite series over dominant weights of G . This could be regarded as an analogue of Witten's volume formula in 2-dimensional gauge theory. Each term includes data such as special values of the characters of the irreducible representations of G associated to the dominant weights.

Singular cotangent bundle reduction & spin Calogero–Moser systems

We develop a bundle picture for singular symplectic quotients of cotangent bundles acted upon by cotangent lifted actions for the case that the configuration manifold is of single orbit type. Furthermore, we give a formula for the reduced symplectic form in this setting. As an application of this bundle picture we consider Calogero–Moser systems with spin associated to polar representations of compact Lie groups.

Kähler quantization and reduction

Exploiting a notion of Kähler structure on a stratified space introduced elsewhere we show that, in the Kähler case, reduction after quantization is equivalent to quantization after reduction in a certain weak sense. Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kähler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization is weakly equivalent to quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond as complex vector spaces (whether or not the natural inner products correspond is unknown whence the qualifier “weakly”) but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kähler spaces arise from representations of compact Lie groups and from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces. For illustration, we carry out Kähler quantization on various spaces of that kind including singular Fock spaces.

Signature quantization and representations of compact Lie groups.

We discuss some applications of signature quantization to the representation theory of compact Lie groups. In particular, we prove signature analogues of the Kostant formula for weight multiplicities and the Steinberg formula for tensor product multiplicities. Using symmetric functions, we also find, for type A, analogues of the Weyl branching rule and the Gel'fand-Tsetlin theorem.

Representation Rings of Classical Groups and Hopf Algebras

We prove a double coset formula for induced representations of compact Lie groups. We apply it to the representation rings of unitary and symplectic groups to obtain Hopf algebras. We also construct a Heisenberg algebra representation based on the restiction and induction of representations of unitary groups.

Cycles of Bott-Samelson type for taut representations

Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut, is either variationally complete or it is one of the following orthogonal representations (n bigger than or equal to 2): the (standard) x_R (spin) representation of SO(2)xSpin(9); or the (standard) x_C (standard) representation of U(2)xSp(n); or the (standard)^3 x_ H (standard) representation of SU(2)xSp(n). In this paper we will show how to adapt the construction of the cycles of Bott and Samelson to the orbits of these three representations. As a result, they also admit explicit cycles representing a basis of their Z_2-homology and, in particular, this provides another proof of their tautness.

Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces

The Leray-Serre spectral sequence Ranks of homotopy groups Some homogeneous spaces Representations of compact Lie groups The case when $G$ is simple The case when $G$ is semisimple Homogeneous compact quadrangles Homogeneous focal manifolds Bibliography.

Lifts of Diffeomorphisms of Orbit Spaces for Representations of Compact Lie Groups

A structure of the group of diffeomorphisms for the orbit space of an orthogonal representation G> →O(V) of a compact Lie group G is studied. For a finite G, it is proved that each diffeomorphism of the orbit space V/G has a smooth lift to V and the component group of the group of Diff(V/G is described, especially in the case when G is a finite Coxeter group. Similar results are obtained for the isotropy representations of some symmetric spaces.

Gelfand pairs attached to representations of compact Lie groups

For each compact Lie algebra g and each real representationV of g we construct a two-step nilpotent Lie groupN(g, V), endowed with a natural left-invariant riemannian metric. The main goal of this paper is to show that this construction produces many new Gelfand pairs associated with nilpotent Lie groups. Indeed, we will give a full classification of the manifoldsN(g, V) which are commutative spaces, using a characterization in terms of multiplicity-free actions.

Lifting smooth curves over invariants for representations of compact Lie groups

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

Symmetric weights and $s$-representations

We study irreducible representations of compact Lie groups relating an algebraic condition (the highest weight λ is i.e., in any simple factor all non zero ζλ, oC) are equal, for any positive root α and any invariant inner product) with a geometnc one (for all orbits, the d-Xh osculating space coincides with the representation space). We prove that, if d — 2 and λ is symmetric, the irreducible representation with highest weight λ corresponds to the isotropy representation of a symmetric space.

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces.

Explicit realization of irreducible representations of classical compact lie groups in the spaces of sections of line bundles

We use the Borel-Weil scheme for the construction of irreducible representations of compact Lie groups in the spaces of holomorphic sections of line bundles over homogeneous manifolds. We find the explicit form of the space of sections and construct an invariant scalar product. We show that the space of holomorphic sections locally satisfies the Zhelobenko indicator system.