Lie Subgroup Research Articles

Classifications of bundle connection pairs by parallel translation and lassos

Let M be a connected manifold and G be a closed Lie subgroup of GL( V)—the general linear group on a finite dimensional vector space V. Denote by ( Ω m ) the space of H 1-loops in M starting at a fixed point ( m). Let M be the set of P ϵ C ∞(Ω m, G) , modulo conjugation by an element of G, such that P( στ) = P( σ) P( τ) for σ, τ ϵ Ω m (στ is the concatenation of the loops σ and τ), P( σ′) = P( σ) if σ′ is reparameterization of σ, and the differential of P satisfies a “locality” condition. It is shown that the bundle connection pairs ( E, ▽) (up to equivalence), with structure group G and fiber V, are in one to one correspondence with M —a similar result has been announced by Kobayashi. The correspondence is induced by the parallel translation operators of connections. Furthermore, if the manifold ( M) is simply connected, then the space of bundle connection pairs can be classified by a collection of Lie algebra valued 1-forms on the manifold Ω m (called integrated lassos). These 1-forms are related to the differentials of elements of M . This last result generalizes Weil's characterization of U(1)—line bundle connection pairs by the curvature 2-form. It is also a generalization of Gross' results to base manifolds ( M) other than R n .

Maximal Homotopy Lie Subgroups of Maximal Rank

Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows: (1) Here π, ϕ, ϕ 1 and ϕ 2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes. Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

Lie symmetries of a generalised nonlinear Schrodinger equation: I. The symmetry group and its subgroups

The symmetry group of the generalised non-linear Schrodinger equation i psi t+ Delta psi =a0 psi +a1 mod psi mod 2 psi +a2 mod psi mod 4 psi in three space dimensions is shown to be the extended Galilei group G(3), for a1a2 not=0, and the Galilei-similitude group Gd(3) (including a dilation) for a1=oor a2=0. All Lie subgroups of G(3) and Gd(3) are found. They will be used in a subsequent paper to obtain group invariant solutions of the equation.

Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups

Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $\operatorname {Ind} (H \uparrow G, \sigma )$. One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraic-geometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over $R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $H$ a complex Lie subgroup, then the multiplicity is a.e. constant.

Complex Algebraic Geometry and Calculation of Multiplicities for Induced Representations of Nilpotent Lie Groups

Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $\operatorname {Ind} (H \uparrow G, \sigma )$. One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraic-geometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over $R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $H$ a complex Lie subgroup, then the multiplicity is a.e. constant.

Direct Integral Decompositions and Multiplicities for Induced Representations of Nilpotent Lie Groups

Let $K$ be a Lie subgroup of the connected, simply connected nilpotent Lie group $G$, and let $\mathfrak {k}$, $\mathfrak {g}$ be the corresponding Lie algebras. Suppose that $\sigma$ is an irreducible unitary representation of $K$. We give an explicit direct integral decomposition of ${\operatorname {Ind} _{k \to G}}\sigma$ into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between $G^\wedge$ and the coadjoint orbits in ${\mathfrak {g}^{\ast }}$ (and similarly for $K^\wedge , {\mathfrak {k}^{\ast }}$). Let $P:{\mathfrak {k}^{\ast }} \to {\mathfrak {g}^{\ast }}$ be the canonical projection, let ${\mathcal {O}_\sigma } \subset {\mathfrak {k}^{\ast }}$ be the orbit corresponding to $\sigma$, and, for $\pi \in G^\wedge$, let ${\mathcal {O}_\pi } \subset {\mathfrak {g}^{\ast }}$ be the corresponding orbit. The main result of the paper says essentially that $\pi \in G^\wedge$ appears in the direct integral iff ${P^{ - 1}}({\mathcal {O}_\sigma })$ meets ${\mathcal {O}_\pi }$; the multiplicity of $\pi$ is the number of ${\operatorname {Ad} ^{\ast }}(K)$-orbits in ${\mathcal {O}_\pi } \cap {P^{ - 1}}({\mathcal {O}_\sigma })$. There is also a natural description of the measure class in the integral.

Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups

Let K K be a Lie subgroup of the connected, simply connected nilpotent Lie group G G , and let k \mathfrak {k} , g \mathfrak {g} be the corresponding Lie algebras. Suppose that σ \sigma is an irreducible unitary representation of K K . We give an explicit direct integral decomposition of Ind k → G σ {\operatorname {Ind} _{k \to G}}\sigma into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between G ∧ G^\wedge and the coadjoint orbits in g ∗ {\mathfrak {g}^{\ast }} (and similarly for K ∧ , k ∗ K^\wedge ,\,{\mathfrak {k}^{\ast }} ). Let P : k ∗ → g ∗ P:{\mathfrak {k}^{\ast }} \to {\mathfrak {g}^{\ast }} be the canonical projection, let O σ ⊂ k ∗ {\mathcal {O}_\sigma } \subset {\mathfrak {k}^{\ast }} be the orbit corresponding to σ \sigma , and, for π ∈ G ∧ \pi \in G^\wedge , let O π ⊂ g ∗ {\mathcal {O}_\pi } \subset {\mathfrak {g}^{\ast }} be the corresponding orbit. The main result of the paper says essentially that π ∈ G ∧ \pi \in G^\wedge appears in the direct integral iff P − 1 ( O σ ) {P^{ - 1}}({\mathcal {O}_\sigma }) meets O π {\mathcal {O}_\pi } ; the multiplicity of π \pi is the number of Ad ∗ ( K ) {\operatorname {Ad} ^{\ast }}(K) -orbits in O π ∩ P − 1 ( O σ ) {\mathcal {O}_\pi } \cap {P^{ - 1}}({\mathcal {O}_\sigma }) . There is also a natural description of the measure class in the integral.

Conformal invariance of white noise

The remarkable link between the structure of the white noise and that of the infinite dimensional rotation group has been exemplified by various approaches in probability theory and harmonic analysis. Such a link naturally becomes more intricate as the dimension of the time-parameter space of the white noise increases. One of the powerful method to illustrate this situation is to observe the structure of certain subgroups of the infinite dimensional rotation group that come from the diffeomorphisms of the time-parameter space, that is the time change. Indeed, those subgroups would shed light on the probabilistic meanings hidden behind the usual formal observations. Moreover, the subgroups often describe the way of dependency for Gaussian random fields formed from the white noise as the time-parameter runs over the basic parameter space.The main purpose of this note is to introduce finite dimensional subgroups of the infinite dimensional rotation group that have important probabilistic meanings and to discuss their roles in probability theory. In particular, we shall see that the conformal invariance of white noise can be described in terms of the conformal group which is a finite dimensional Lie subgroup of the infinite dimensional rotation group.

Markov-type Lie groups in GL(n,R)

The general linear group GL(n,R) is decomposed into a Markov-type Lie group and an abelian scale group. The Markov-type Lie group basis is shown to generate all singly stochastic matrices which are continuously connected to the identity when non-negative parameters are used. A basis is found which shows that it in turn contains a Lie subgroup which corresponds to doubly stochastic matrices, which basis, over the complex field, is shown to give the symmetric group for certain discrete values of the complex parameters. The basis of the Markov algebra is shown to give the negative of the corresponding M-matrices with property ‘‘C’’ (for non-negative combinations). These stochastic Lie groups are shown to be isomorphic to the affine group and the general linear group in one less dimension. The basis generates transformations with a natural interpretation for physical applications.

Polar coordinates induced by actions of compact Lie groups

Let G G be a connected Lie subgroup of the real orthogonal group O ( n ) O(n) . For the action of G G on R n {{\mathbf {R}}^n} , we construct linear subspaces a \mathfrak {a} that intersect all orbits. We determine for which G G there exists such an a \mathfrak {a} meeting all the G G -orbits orthogonally; groups that act transitively on spheres are obvious examples. With few exceptions all possible G G arise as the isotropy subgroups of Riemannian symmetric spaces.

Compact Lie group action and equivariant bordism

Let G G be a compact Lie group and H H a compact Lie subgroup of G G contained in the center of G G with H m {H^m} the maximal subgroup in the center, H H being H H -boundary. Let p r : H m → H pr:{H^m} \to H be the projection onto the r r th factor and H r {H_r} be the r r th factor of H m {H^m} . Let { L r } \{ {L_r}\} be a family of subgroups of G G such that L r ∩ H r {L_r} \cap {H_r} is nontrivial. Consider a G G -manifold M n {M^n} with p r ( G x ∩ H m ) pr({G_x} \cap {H^m}) trivial or containing L r {L_r} , for every x x in M n {M^n} . The main result of the paper is that if ∀ x ∈ M n \forall x \in {M^n} , p r ( G x ∩ H m ) {p_r}({G_x} \cap {H^m}) is trivial at least for one r r , then M n {M^n} is a G G -boundary.

On The Lie Group of Umbrella Matrices

In this paper, Umbrella matrices are defined which are selected from GL (n, |^ ) and it i» shown that Umbrella matrices form a matrix group with respect to the matrix product. Next, a characterization of Umbrella matrices is given; among these it İs proved that the gro­ up of Umbrella matices which are selected from O (n) and the group of Doubly Umbrella matrices are subgroup of Umbrella matrices which are selected from GL (n, Later it is shown that this matrix group is a Lie subgroup of GL (n, and a characteristic property is given about this Lie group. Finally it is shown that, the Lie group under consideration is not connected and noncompact are shown. At the end of this work, the Maurer-Cartan forms are investigated and using the prin- cipal I-forms the dimension of the Lie group under consideration is computed.

The holonomy group and spontaneous symmetry breaking

We show that the residual symmetry group after spontaneous symmetry breaking must either be the holonomy group H of the connection or contain H as a subgroup. Since H is a connected, normal Lie subgroup of the gauge group G, knowing the dimension of H is often sufficient to determine H itself. We develop an algorithm for determining dim H for any gauge theory given the structure constants of the gauge group. This algorithm is then applied to a fiber bundle with an SU2 or SU3 gauge group.

Decomposition of elements of the automorphism group of a torus action, I

Let (~ be a torus acting smoothly (C °') and effectively on the compact connected C ~ manifold X. Let Er(X, G) be the group of equivariant dfffeomorphisms of class C r of X onto X such that each orbit is an invariant subset. If all isotropy subgroups are finite it is known that Er(x, G) can be given the structure of a Banach Lie group (1 ~ r ~ o0) in the following way. Given [ in Er(X, G) there is a unique C r map • : X -~ G such that [(x) = ~(x)x for all x in X. The correspondence of ~ with [ allows identification of Er(X, G) with a Banach Lie subgroup of the Banach Lie group Ct(X, (7) [6]. In this paper we show that the finiteness of all isotropy subgroups is necessary condition. Indeed, if some point in X has an infinite isotropy subgroup, then for each nonnegative integer r there exists an element [r of Er+I(X,G) and an element ~r of Cr(X,G)- cr+I(X,G) such that It(x)-~ ~r(X)X for all x in X. Before proceeding with the proof of our result, we require the following computational proposition, which is proven in [7].

A unification of boson expansion theories: (II). Boson expansions as provided by the functional representation method

It is shown that the boson expansions hitherto known, as well as an infinite number of the new ones, can be derived in a unified way in terms of the functional representation technique. Each boson expansion (boson representation) is valid for the carrier space of an irreducible representation of a semisimple compact Lie subgroup of SO(2 N + 1) and can be presented as Dyson-type, Holstein-Primakoff-type or Garbaczewski (Marumori)-type expansion with the corresponding properties concerning finiteness, convergence and hermitian conjugation. The physical boson space can be determined for every expansion and the projection operator is explicitly found. The methods for solving the Schrödinger equation are discussed and the generator coordinate method is shown to be equivalent to the boson expansion approach.

A Classification of Homogeneous Surfaces

Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.

A note on coherent states

Given a connected Lie groupG with an Abelian invariant Lie subgroup and a continuous unitary representation ofG on the Hilbert space ℋ, we investigate a relationship between the first cohomology groupH1(G, ℋ) and classes of sectors, determined by coherent states with a projectivelyG-covariant Weyl system. This result is applied to calculateH1(G, ℋ), if the groupG has in addition a compact subgroup with certain properties.

A symmetric space of noncompact type has no equivariant isometric immersions into the Euclidean space

PROOF. Replacing G by a finite covering we may and will assume that G is a direct product of simple Lie groups, none of which is compact. The group In of isometries of the Euclidean n-space is known to be the semidirect product of the orthogonal group and R , RI being the normal subgroup. Therefore the projection of In onto the orthogonal group is a differentiable group morphism. Let f be a differentiable morphism from G into In. Then f followed by the projection of In onto the orthogonal group is a differentiable morphism. If this composition were nontrivial it would be nontrivial on some simple factor of G. Therefore, there would be a noncompact simple Lie group contained in the orthogonal group. This is impossible because a semisimple Lie subgroup of a compact Lie group is closed, hence compact (cf. [2, p. 128D. Thus, the image of f is contained in RW. This implies that G has a nontrivial abelian factor, unless f is the trivial morphism.

Homologies and elations in compact, connected projective planes

In a compact, connected topological projective plane, let Ω be a closed Lie subgroup of the group of all axial collineations with a fixed axis A. We compare the set З\\ A consisting of the centres of all non-identical homologies in Ω to orbits of the group Ω [A] of all elations contained in Ω and of its connected component θ = (Ω [A]) 1 . It is shown that З\\ A is the union of at most countably many θ-orbits; moreover, З\\ A turns out to be a single θ-orbit whenever the connected component of Ω contains non-identical homologies. This result is analogous to a well-known theorem of André for finite planes. It has numerous consequences for the structure of collineation groups of compact, connected projective planes.

Kronecker products of representations of exceptional Lie groups

A technique is described for evaluating Kronecker products of irreducible representations of an exceptional Lie group, G. The method depends upon the natural embedding in G of a classical Lie subgroup H. The problem is reduced to that of finding the branching of one irreducible representation of G on restriction to H, evaluating Kronecker products in H and finally using the modification rules appropriate to G.