Noncompact Lie Group Research Articles

On the volume of orbifold quotients of symmetric spaces

Key to H. C. Wang's quantitative study of Zassenhaus neighbourhoods of non-compact semisimple Lie groups are two constants that depend on the root system of the corresponding Lie algebra. This article extends the list of values for Wang's constants to the exceptional Lie groups and also removes their dependence on dimension. The first application is an improved upper sectional curvature bound for a canonical left-invariant metric on a semisimple Lie group. The second application is an explicit uniform positive lower bound for arbitrary orbifold quotients of a given irreducible symmetric space of non-compact type.

Explicit ambient metrics and holonomy

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit Fefferman-Graham ambient metrics provide a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\mathbf{G}_2$ as well as ambient metrics with holonomy contained in $\mathbf{Spin}(4,3)$.

Homogeneous almost-Kähler manifolds and the Chern–Einstein equation

Given a non-compact semisimple Lie group G we describe all homogeneous spaces G / L carrying an invariant almost-Kahler structure $$(\omega ,J)$$ . When L is abelian and G is of classical type, we classify all such spaces which are Chern–Einstein, i.e. which satisfy $$\rho = \lambda \omega $$ for some $$\lambda \in {\mathbb {R}}$$ , where $$\rho $$ is the Ricci form associated to the Chern connection.

Local Type II metrics with holonomy in $${\mathrm {G}}_2^*$$

A list of possible holonomy groups with indecomposable holonomy representation contained in the exceptional, non-compact Lie group $${\mathrm {G}}_2^{*}$$ was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1, 2 or 3, respectively. It was also shown by Fino and Kath that all algebras of Type I, and by the author that all of Type III are indeed realizable as holonomy algebras by metrics with signature (4, 3). This article proves that this is also true for all Type II algebras. Thus, there exists a realization by a metric for all indecomposable holonomy groups contained in $${\mathrm {G}}_2^{*}$$.

SU(2) and SU(1,1) group parametrizations via qubit induced correlated probabilities

A group is an algebraic structure equipped with binary associative multiplication, identity element and inverse operation with respect to identity. Lie group is a smooth manifold with group multiplication and the inverse being smooth functions. Therefore as one chooses manifold coordinate charts, various parametrizations of multiplication and inverse operations arise. Recently, the simple probability representation of qubit states and observables and an associated triangle geometrical picture called ‘quantum suprematism’ were introduced. At its core lies the representation of the Bloch sphere by correlated dichotomic probability distribution. The aim of the present paper is the introduction of probability parametrization of two classical compact and non-compact Lie groups, SU(2) and SU(1,1) respectively.

De Rham 2-Cohomology of Real Flag Manifolds

Let $\mathbb{F}_{\Theta }=G/P_{\Theta }$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $% P_{\Theta }$. This is a closed subgroup of $G$ determined by a subset $% \Theta $ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_\Theta$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_\Theta,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic classes of principal fiber bundles. The starting point is the computation of the second homology group of $\mathbb{F}_{\Theta }$ with coefficients in a ring $R$.

Matrix logarithms and range of the exponential maps for the symmetry groups , and the Lorentz group

Physicists know that covering the continuously connected component of the Lorentz group can be achieved through two Lie algebra exponentials, whereas one exponential is sufficient for compact symmetry groups like SU(N) or SO(N). On the other hand, both the general Baker-Campbell-Hausdorff formula for the combination of matrix exponentials in a series of higher order commutators, and the possibility to define the logarithm of a general matrix through the Jordan normal form, seem to naively suggest that even for non-compact groups a single exponential should be sufficient. We provide explicit constructions of for all matrices in the fundamental representations of the non-compact groups , and SO(1, 2). The construction for also yields logarithms for SO(1, 3) through the spinor representations. However, it is well known that single Lie algebra exponentials are not sufficient to cover the Lie groups and . Therefore we revisit the maximal neighbourhoods and which can be covered through single exponentials with or , respectively, to clarify why or outside of the corresponding domains or . On the other hand, for the Lorentz groups SO(1, 2) and SO(1, 3), we confirm through construction of the logarithm that every transformation in the connectivity component of the identity element can be represented in the form with or , respectively. We also examine why the proper orthochronous Lorentz group can be covered by single Lie algebra exponentials, whereas this property does not hold for its covering group : The logarithms in correspond to logarithms on the first sheet of the covering map , which is contained in . The special linear groups and the Lorentz group therefore provide instructive examples for different global behaviour of non-compact Lie groups under the exponential map.

Strichartz estimates for the metaplectic representation

We provide new estimates for the matrix coefficients of the metaplectic representation, inspired by a formal analogy with the Strichartz estimates which hold for several classes of evolution propagators U(t) . The one parameter group of unitary operators U(t) is replaced by a unitary representation of a non-compact Lie group, the group element playing the role of time; the case of the metaplectic or oscillatory representation is of special interest in this connection, because the Schrödinger group is a subgroup of the metaplectic group. We prove uniform weak-type sharp estimates for matrix coefficients and Strichartz-type estimates for that representation. The crucial point is the choice of function spaces able to detect such a decay, which in general will depend on the given group action. The relevant function spaces here turn out to be the so-called modulation spaces from time-frequency analysis in Euclidean space, and Lebesgue spaces with respect to Haar measure on the metaplectic group. The proofs make use in an essential way of the covariance of the Wigner distribution with respect to the metaplectic representation.

A multiplier theorem for sub-Laplacians with drift on Lie groups

We prove a general multiplier theorem for symmetric left-invariant sub-Laplacians with drift on non-compact Lie groups. This considerably improves and extends a result by Hebisch, Mauceri, and Meda. Applications include groups of polynomial growth and solvable extensions of stratified groups.

GEOMETRIC CYCLES IN COMPACT LOCALLY HERMITIAN SYMMETRIC SPACES AND AUTOMORPHIC REPRESENTATIONS

Let G be a linear connected non-compact real simple Lie group and let K ⊂ G be a maximal compact subgroup of G. Suppose that the centre of K is isomorphic to 𝕊1 so that X := G/K is a global Hermitian symmetric space. Denote the compact dual of X by Xu. Let θ be the Cartan involution of G that fixes K. Let Γ be a uniform arithmetic lattice in G such that θ(Γ) = Γ. Suppose that G is one of the groups SU(p, q), p < q – 1, q ≥ 5, SO0(2, q), Sp(n, ℝ), n ≠ 4, SO*(2n), n ≥ 9. Then there exists a unique irreducible unitary representation $$ {\mathcal{A}}_{\mathfrak{q}} $$ associated to a proper θ-stable parabolic subalgebra $$ \mathfrak{q} $$ with R+( $$ \mathfrak{q} $$ ) = R−( $$ \mathfrak{q} $$ ) such that if Hs,s( $$ \mathfrak{g} $$ , K; $$ {A}_{\mathfrak{q}\prime, K} $$ ) ≠ 0 for some 0 < s ≤ R+( $$ \mathfrak{q} $$ ), then $$ {\mathcal{A}}_{\mathfrak{q}\prime } $$ is unitarily equivalent to either the trivial representation or to $$ {\mathcal{A}}_{\mathfrak{q}} $$ . As a consequence, using certain geometric cycles C ⊂ XΓ := Γ\X constructed by Millson and Raghunathan, we show that under suitable hypotheses on Γ, for any sublattice Λ ⊂ Γ, the representation $$ {\mathcal{A}}_{\mathfrak{q}} $$ occurs in L2(Λ\G) with positive multiplicity.

Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple lie groups

Let G be a non-compact semisimple Lie group with finite centre and finitely many connected components. We show that any finitely generated group Γ which is quasi-isometric to an irreducible lattice in G has the R∞-property, namely, that there are infinitely many ϕ-twisted conjugacy classes for every automorphism ϕ of Γ. Also, we show that any lattice in G has the R∞-property, extending our earlier result for irreducible lattices.

Cellular homology of real flag manifolds

Let FΘ=G∕PΘ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and PΘ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow FΘ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in Rn and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ±2 (so that Z2-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.

Communicating without shared reference frames

We generalize a quantum communication protocol introduced by Bartlett et al. [New J. Phys. 11, 063013 (2009)], in which two parties communicating do not share a classical reference frame, to the case where changes of their reference frames form a one-dimensional noncompact Lie group. Alice sends to Bob the state ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$, where ${\ensuremath{\rho}}_{S}$ is the state of the system Alice wishes to communicate and ${\ensuremath{\rho}}_{R}$ is the state of an ancillary system serving as a token of her reference frame. Because Bob is ignorant of the relationship between his reference frame and Alice's, he will describe the state ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$ as an average over all possible reference frames. Bob measures the reference token and applies a correction to the system Alice wished to communicate conditioned on the outcome of the measurement. The recovered state ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ is decohered with respect to ${\ensuremath{\rho}}_{S}$, the amount of decoherence depending on the properties of the reference token ${\ensuremath{\rho}}_{R}$. We present an example of this protocol when Alice and Bob do not share a reference frame associated with the one-dimensional translation group and use the fidelity between ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ to quantify the success of the recovery operation.

Characteristic functions of semigroups in semi-simple Lie groups

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G = K ⁢ A ⁢ N {G=KAN} . For a semigroup S ⊂ G {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform I S ⁢ ( λ , u ) = ∫ S e λ ⁢ ( 𝖺 ⁢ ( g , u ) ) ⁢ 𝑑 g {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, u ∈ K {u\in K} , λ ∈ 𝔞 ∗ {\lambda\in\mathfrak{a}^{\ast}} , 𝔞 {\mathfrak{a}} is the Lie algebra of A and g ⁢ u = k ⁢ e 𝖺 ⁢ ( g , u ) ⁢ n ∈ K ⁢ A ⁢ N {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written 𝔽 Θ ⁢ ( S ) {\mathbb{F}_{\Theta(S)}} called the flag type of S, where Θ ⁢ ( S ) {\Theta(S)} is a subset of the simple system of roots. It is proved that I S ⁢ ( λ , u ) &lt; ∞ {I_{S}(\lambda,u)&lt;\infty} if λ belongs to a convex cone defined from Θ ⁢ ( S ) {\Theta(S)} and u ∈ π - 1 ⁢ ( 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ) {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ⊂ 𝔽 Θ ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and π : K → 𝔽 Θ ⁢ ( S ) {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of I S ⁢ ( λ , u ) {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in 𝒟 Θ ⁢ ( S ) ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)} invariant by the group S ∩ S - 1 {S\cap S^{-1}} of units of S.

Expanding Cone and Applications to Homogeneous Dynamics

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

Branching laws: some results and new examples

For a connected, noncompact simple matrix Lie group G so that a maximal compact subgroup K has a three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contain the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general G such that G/K is an Hermitian G-manifold we give a necessary and sufficient condition so that an arbitrary square integrable representation of the ambient group is admissible over the semisimple factor of K.

String structures associated to indefinite Lie groups

String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group O(n) to the indefinite-signature orthogonal group O(p,q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simpler, constructions hold for other noncompact Lie groups such as the unitary group U(p,q) and the symplectic group Sp(p,q). This extension provides a starting point for an abundance of constructions in (higher) geometry and applications in physics.

Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford–Klein forms problem

In this article, we develop the algorithm for finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford–Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford–Klein forms. By a numerical experiment, we show that there is a large class of homogeneous spaces satisfying Tholozan’s condition.

Generalized Flow-Box property for singular foliations

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety X (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (Invent Math 103(2):297–325, 1991, Theorem 3.2.1) about the existence of local slices for a (non-compact) Lie group action.

On density of infinite subsets II: Dynamics on homogeneous spaces

Let $G$ be a noncompact semisimple Lie group, $\Gamma$ be an irreducible cocompact lattice in $G$, and $P<G$ be a minimal parabolic subgroup. We consider the dynamics of $P$ acting on $G/\Gamma$ by left translation. For any infinite subset $A\subset G/\Gamma$, we show that, for any $\epsilon>0$, there is a $g\in P$ such that $gA$ is $\epsilon$-dense. We also prove a similar result for certain discrete group actions on $\mathbb T^n$.