Compact Semisimple Lie Algebra Research Articles

A note on commutators in compact semisimple Lie algebras

A note on commutators in compact semisimple Lie algebras

Canonical domains for coadjoint orbits

AbstractThis paper describes two real analytic symplectomorphisms defined on appropriate dense open subsets of any coadjoint orbit of a compact semisimple Lie algebra. The first symplectomorphism sends the open dense subset to a bounded subset of a standard cotangent bundle. The second symplectomorphism has as target a bounded subset of a hyperbolic coadjoint orbit of an associated non‐compact semisimple Lie algebra. Therefore, coadjoint orbits of compact Lie algebras are symplectic compactifications of domains of cotangent bundles, and are in symplectic correspondence with hyperbolic orbits of non‐compact semisimple Lie algebras.

Box splines, tensor product multiplicities and the volume function

We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra 𝔤 and a special function 𝒥 associated to 𝔤, called the volume function. The volume function arises in connection with the randomized Horn’s problem in random matrix theory and has a related significance in symplectic geometry. Building on box spline deconvolution formulae of Dahmen–Micchelli and De Concini–Procesi–Vergne, we develop new techniques for computing the multiplicities from 𝒥, answering a question posed by Coquereaux and Zuber. In particular, we derive an explicit algebraic formula for a large class of Littlewood–Richardson coefficients in terms of 𝒥. We also give analogous results for weight multiplicities, and we show a number of further identities relating the tensor product multiplicities, the volume function and the box spline. To illustrate these ideas, we give new proofs of some known theorems.

Ribbon Braided Module Categories, Quantum Symmetric Pairs and Knizhnik–Zamolodchikov Equations

Let $\mathfrak u$ be a compact semisimple Lie algebra, and $\sigma$ be a Lie algebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbon braided tensor C*-category of $U_q(\mathfrak u)$-representations for $0<q<1$. We introduce three module C*-categories over Rep$_q(\mathfrak u)$ starting from the input data $(\mathfrak u,\sigma)$. The first construction is based on the theory of cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module C*-category is ribbon twist-braided in the sense of A. Brochier---this is essentially due to B. Enriquez in the first case, is proved by S. Kolb in the second case, and is closely related to work of J. Donin, P. Kulish, and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these ribbon twist-braided module C*-categories, and confirm it in the rank one case.

On squares of representations of compact Lie algebras

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.

Deformations of the Lie–Poisson sphere of a compact semisimple Lie algebra

AbstractA compact semisimple Lie algebra $\mathfrak{g}$ induces a Poisson structure $\pi _{\mathbb{S}}$ on the unit sphere $\mathbb{S}(\mathfrak{g}^*)$ in $\mathfrak{g}^*$. We compute the moduli space of Poisson structures on $\mathbb{S}(\mathfrak{g}^*)$ around $\pi _{\mathbb{S}}$. This is the first explicit computation of a Poisson moduli space in dimension greater or equal than three around a degenerate (i.e. not symplectic) Poisson structure.

Rigidity around Poisson submanifolds

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem; another interesting case corresponds to spheres inside duals of compact semisimple Lie algebras, our result can be used to fully compute the resulting Poisson moduli space.

Closed geodesics on compact nilmanifolds with Chevalley rational structure

We study the distribution of closed geodesics on nilmanifolds Γ \ N arising from a 2-step nilpotent Lie algebra \({\mathfrak {N}}\) constructed from an irreducible representation of a compact semisimple Lie algebra \({\mathfrak {g}o}\) on a real finite dimensional vector space U. We determine sufficient conditions on the semisimple Lie algebra \({\mathfrak {g}o}\) for Γ \ N to have the density of closed geodesics property where Γ is a lattice arising from a Chevalley rational structure on \({\mathfrak{N}}\) .

TWO-STEP NILPOTENT LIE ALGEBRAS WITH AD-INVARIANT METRICS AND A SPECIAL KIND OF SKEW-SYMMETRIC MAPS

We prove that a 2-step nilpotent Lie algebra admitting an ad-invariant metric can be constructed from a vector space 𝔳 endowed with an inner product 〈,〉 and an injective homomorphism ρ : 𝔳 → 𝔰𝔬(𝔳) satisfying ρ(v)v = 0 for all v ∈ 𝔳. The corresponding simply connected pseudo-Riemannian Lie groups are flat and any isometry fixing the identity element does not depend on ρ. The description allows one to construct examples starting with a compact semisimple Lie algebra and it is useful to show some applications.

On non-Abelian expanding waves

We study non-Abelian expanding waves that can be radiated from various sources of Yang–Mills fields. We find a new class of exact wave solutions to the Yang–Mills equations. These solutions are constructed for any gauge group with a compact semi-simple Lie algebra and embrace asymmetrical cases of radiations of expanding waves. They can be regarded as a reasonable generalization of wave solutions of the Maxwell electrodynamics. It is of interest to apply the found solutions to detect cosmic sources of Yang–Mills fields. In the case of fields with SU(2) symmetry this could be realized by observing the interaction of such sources' radiation with neutrinos.

Constructing the classical limit for quantum systems on compact semisimple Lie algebras

We give a general construction for the classical limit of a quantum system defined in terms of generators of an arbitrary compact semisimple Lie algebra, generalizing known results for the $\mathfrak{su}_2$ and $\mathfrak{su}_3$ cases. The classical limit depends on the physical problem in question and is determined by the sequence of representations by which it is reached. Only in the simplest cases it is unique. We present explicit formulae useful in determining the classical limit in all important cases.

Programs for generating Clebsch–Gordan coefficients of SU(3) in SU(2) and SO(3) bases

Computer codes are developed to calculate Clebsch–Gordan coefficients of SU(3) in both SU(2)- and SO(3)-coupled bases. The efficiency of this code derives from the use of vector coherent state theory to evaluate the required coefficients directly without recursion relations. The approach extends to other compact semi-simple Lie groups. The codes are given in subroutine form so that users can incorporate the codes into other programs. Program summary Title of program: SU3CGVCS Catalogue identifier: ADTN Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTN Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: Persons requesting the program must sign the standard CPC non-profit use license Computers for which the program is designed and others on which it is operable: SGI Origin 2000, HP Apollo 9000, Sun, IBM SP, Pentium Operating systems under which the program has been tested: IRIX 6.5, HP UX 10.01, SunOS, AIX, Linux Programming language used: FORTRAN 77 Memory required to execute with typical data: On the HP system, it requires about 732 KBytes. Disk space used for output: 2100+2460 bytes No. of bits in a word: 32 bit integer and 64 bit floating point numbers. No. of processors used: 1 Has the code been vectorized: No No. of bytes in distributed program, including test data, etc.: 26 309 No. of lines in distributed program, including test data, etc.: 3969 Distribution format: tar gzip file Nature of physical problem: The group SU(3) and its Lie algebra su(3) have important applications, for example, in elementary particle physics, nuclear physics, and quantum optics [1–3]. The code presented is particularly relevant for the last two fields. Clebsch–Gordan (CG) coefficients are required whenever the symmetries of many-body systems are used for the evaluation of matrix elements of tensor operators. Moreover, the construction of CG coefficients for SU(3) serves as a nontrivial prototype for larger compact semi-simple Lie algebras and even for non semi-simple Lie algebras. It is the simplest Lie algebra to have multiplicity in its outer products and a non-canonical subalgebra, i.e., SO(3). Method of solution: Vector coherent state theory is first used to construct bases for the products of two irreducible representations (irreps) [4]. The bases are SU(2)-coupled so that SU(2)-reduced CG (or isoscalar factors) can be constructed naturally. The CG coefficients in the SO(3) bases are constructed subsequently from the overlaps between the SU(2) and SO(3) bases. Restriction on the complexity of the problem: The programs are limited by computer memory and the maximum size of variable arrays. As dimension overflow conditions are possible, they are flagged and can be fixed by following the directions given as part of the error message. Typical running time: The calculation time for a single SU(3) CG coefficient is very different for SU(2) and SO(3) bases. It varies between 7.3–54.1 ns in SGI Origin 2000, 0.81–5.48 ms in HP Apollo 9000, or 0.055–0.373 ms in Intel Pentium 4 for SU(2) bases while it is between 0.027–0.255 s in Intel Pentium 4 for SO(3) bases. Unusual features of the program: Intrinsic bit functions: and, or, and shift, called iand, ior, and ishft, respectively, in FORTRAN, are used for packing and unpacking the labels for the irreps. Intrinsic logical btest is used to test the bit for the phase factor.

Geometry of translations of invariants on semisimple Lie algebras

Each orbit of the coadjoint representation of a semisimple Lie algebra can be equipped with a complete commutative family of polynomials; this family was obtained by the argument-translation method in papers of Mishchenko and Fomenko. This commutative family and the corresponding Euler's equations play an important role in the theory of finite-dimensional integrable systems. These Euler's equations admit a natural Lax representation with spectral parameter. It is proved in the paper that the discriminant of the spectral curve coincides completely with the bifurcation diagram of the moment map for the algebra . The maximal degeneracy points of the moment map are described for compact semisimple Lie algebras in terms of the root structure. It is also proved that the set of regular points of the moment map is connected, and the inverse image of each regular point consists of precisely one Liouville torus.

Cotangent bundles of 4-dimensional hypercomplex Lie groups

We study geometrical structures on the cotangent bundle T * G of a Lie group G which are left-invariant with respect to the Lie group structure on T * G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-Kahler metrics and hyper-Kahler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying Inonu-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures.

On all possible static spherically symmetric EYM solitons and black holes

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein–Yang–Mills equations for any action of the rotation group (or SU(2)) by automorphisms of a principal bundle over spacetime whose structure group is a compact semi-simple Lie group G. These actions are characterized by a vector in the Cartan subalgebra of \U0001d524 and are called regular if the vector lies in the interior of a Weyl chamber. In the irregular cases (the majority for larger gauge groups) the boundary value problem that results for possible asymptotically flat soliton or black hole solutions is more complicated than in the previously discussed regular cases. In particular, there is no longer a gauge choice possible in general so that the Yang–Mills potential can be given by just real-valued functions. We prove the local existence of regular solutions near the singularities of the system at the centre, the black hole horizon, and at infinity, establish the parameters that characterize these local solutions, and discuss the set of possible actions and the numerical methods necessary to search for global solutions. That some special global solutions exist is easily derived from the fact that \U0001d530\U0001d532(2) is a subalgebra of any compact semi-simple Lie algebra. But the set of less trivial global solutions remains to be explored.

Hyper-lie poisson structures

The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10], [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm's equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of sl(2,).

A note on the reality of tensor products

In this paper we prove a statement about the real and pseudoreal (i.e. quaternionic) content of tensor products between real and/or quaternionic representations of compact semisimple Lie algebras (or connected compact semisimple Lie groups) that might be useful in model building when constructing unified theories of fundamental interactions. We also stress the utility of anti-involutions in the description of reality properties.

Representations of compact semisimple Lie algebras over Lagrangian submanifolds

Representations of compact semisimple Lie algebras over Lagrangian submanifolds

Compact simple Lie algebras with two involutions and submanifolds of compact symmetric spaces. II

Introduction. This is a continuation of Part I, which appears in the same Journal. In the previous paper we take a Grassmann bundle GS(TM) over a compact simply connected irreducible riemannian symmetric space M and consider a G-orbit CV in GS(TM) by the isometry group G of M. For each Q? we can define a class of submanifolds in M, so is called, a ^-geometry. We moreover assume that ^ is a G-orbit which contains an ^-dimensional strongly curvature-invariant subspace. Then ^l? corresponds to a PSLA (g, σ, T) of compact semisimple Lie algebra g and two commutative involutions σ, r. PSLA's are algebraicaly divided into those of inner type and those of outer type. Our aim in this article is to prove the following

Aspects of affine Toda field theory

This paper describes affine Toda field theory which is a theory of r scalar fields in two-dimensional Minkowski space-time, where r is the rank of a compact semi-simple Lie algebra g. The classical field theory is determined by the lagrangian density L = 1/2 {partial derivative}{sub {rho}}{phi}{sup a}{partial derivative}{sup {mu}}{phi}{sup a} {minus} V({phi}) where V({phi}) = m{sup 2}/{beta}{sup 2} {Sigma}{sub 0}{sup r}n{sub i}e{sup {beta}{alpha}{sub i} {center dot} {phi}}. m and {beta} are real, classically unimportant constants, {alpha}{sub i} i = 1, . . . ,r are the simple roots of the Lie algebra g, and {alpha}{sub 0} = {Sigma}{sub 1}{sup 4} n{alpha}{sub i} is a linear combination of the simple roots; it corresponds to the extra spot on an extended Dynkin diagram for g. A reasonable question to ask is whether the classical integrability survives into the quantum field theory and, if so, what is the spectrum and to what extent is it possible to calculate explicitly quantities of interest such as S-matrices and form factors. The recent discoveries leave no doubt that these relatively simple models have much structure and their study (even in the {beta}{sup 2} {gt} 0 regime) will be informative. In this short review, the ADEmore » series of Lie algebras will be singled out for special attention.« less