Compact Lie Group Research Articles

Convolution Algebras of Double Groupoids and Strict 2-Groups

Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally, strict 2-groups. Groupoid structures give rise to convolution operations on the space of arrows. Therefore, a double groupoid comes equipped with two product operations on the space of functions. In this article we investigate in what sense these two convolution operations are compatible. We use the representation theory of compact Lie groups to get insight into a certain class of 2-groups.

Estimates for Sums of Eigenfunctions of Elliptic Pseudo-differential Operators on Compact Lie Groups

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

Quantifying noncovariance of quantum channels with respect to groups

A quantum channel is covariant with respect to a group if it commutes with the action of the group. In general, a quantum channel may not be covariant with respect to a given group. The degree of noncovariance can vary between different channels, and it is desirable to have a quantitative characterization for the degree of channel noncovariance. In this work, we propose a measure based on the Hilbert-Schmidt norm to quantify noncovariance of quantum channels with respect to a group and demonstrate that it satisfies several desirable properties. Compared with the existing measures of channel noncovariance, our measure applies to not only compact Lie groups but also finite groups, and it is easy to evaluate. Using this measure and its modified version together with two existing measures, we evaluate and analyze channel noncovariance through an example, finding that these measures of channel noncovariance are closely related but differ from each other. They capture different perspectives of noncovariance of quantum channels. As applications, we provide a relation between channel noncovariance and approximate quantum error correction using our measures of channel noncovariance.

Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair (G,Γ), and we prove that the last coincides with the Lott L2 analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case G=H, the Heisenberg group.

Characterizing barren plateaus in quantum ansätze with the adjoint representation

Variational quantum algorithms, a popular heuristic for near-term quantum computers, utilize parameterized quantum circuits which naturally express Lie groups. It has been postulated that many properties of variational quantum algorithms can be understood by studying their corresponding groups, chief among them the presence of vanishing gradients or barren plateaus, but a theoretical derivation has been lacking. Using tools from the representation theory of compact Lie groups, we formulate a theory of barren plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra, covering a large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks. Our theory provides, for the first time, the ability to compute the exact variance of the gradient of the cost function of the quantum compound ansatz, under mixing conditions that we prove are commonplace.

On gauge Poisson brackets with prescribed symmetry

Abstract In the context of averaging method, we describe a reconstruction of invariant connection-dependent Poisson structures from canonical actions of compact Lie groups on fibered phase spaces. Some symmetry properties of Wong's type equations are derived from the main results

A fixed-point formula for Dirac operators on Lie groupoids

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.

A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws

E. Breuillard showed that finite subsets F\hspace{-.75pt} of matrices in \mathrm{GL}_{d\hspace{-.75pt}}(\hspace{-.75pt}\overline{\mathbb{Q}}\hspace{-.75pt}) generating non-virtually solvable groups have normalized height \hat{h}(F)\ge\varepsilon_d , for some positive \varepsilon_{d}>0 . The normalized height \hat{h}(F) is a measure of the arithmetic size of F and this result can be thought of as a non-abelian analog of Lehmer’s Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.

The nucleus of a compact Lie group, and support of singularity categories

In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite groups.

Equivariant Topological T-Duality

Topological T-duality is a relationship between pairs (E, P) over a fixed space X, where E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document} is a principal torus bundle and P→E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \\rightarrow E$$\\end{document} is a twist, such as a gerbe for a principal PU(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PU({\\mathcal {H}})$$\\end{document}-bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P) and (E^,P^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{E}}, {\\hat{P}})$$\\end{document} comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of E^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{E}}$$\\end{document}. We formulate topological T-duality for circle bundles in the equivariant setting, following the definition of Bunke, Rumpf, and Schick. We define the T-duality transformation in equivariant K-theory and show that it is an isomorphism for all compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g\geq 3$.

A crossed module representation of a 2-group constructed from the 3-loop group Ω3G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega ^3G$$\\end{document}

The quantization of chiral fermions on a 3-manifold in an external gauge potential is known to lead to an abelian extension of the gauge group. In this article, we concentrate on the case of Ω3G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega ^3 G$$\\end{document} of based smooth maps on a 3-sphere taking values in a compact Lie group G. There is a crossed module constructed from an abelian extension Ω3G^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\Omega ^3 G}$$\\end{document} of this group and a group of automorphisms acting on it as explained in a recent article by Mickelsson and Niemimäki. We shall construct a representation of this crossed module in terms of a representation of Ω3G^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\Omega ^3 G}$$\\end{document} on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of Ω3G^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\Omega ^3 G}$$\\end{document} as outer automorphisms of the canonical anticommutation relations algebra in the Fock space.

A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groups

AbstractWe study a Hilbert–Mumford criterion for polystablility associated with an action of a real reductive Lie group G on a real submanifold X of a Kähler manifold Z. Suppose the action of a compact Lie group with Lie algebra $$\mathfrak {u}$$ u extends holomorphically to an action of the complexified group $$U^{\mathbb {C}}$$ U C and that the U-action on Z is Hamiltonian. If $$G\subset U^{\mathbb {C}}$$ G ⊂ U C is compatible, there is a corresponding gradient map $$\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}$$ μ p : X → p , where $$\mathfrak {g}= \mathfrak {k}\oplus \mathfrak {p}$$ g = k ⊕ p is a Cartan decomposition of the Lie algebra of G. Under some mild restrictions on the G-action on X, we characterize which G-orbits in X intersect $$\mu _\mathfrak {p}^{-1}(0)$$ μ p - 1 ( 0 ) in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity ($$\partial _\infty G/K$$ ∂ ∞ G / K ) of the symmetric space G/K. We also establish the Hilbert–Mumford criterion for polystability of the action of G on measures.

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) Lp-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d+-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

Supersymmetry and trace formulas. Part I. Compact Lie groups

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.

TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES

AbstractFor closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$ . Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$ . Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

The five gradients inequality on differentiable manifolds

The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds.

Subelliptic sharp Gårding inequality on compact Lie groups

Subelliptic sharp Gårding inequality on compact Lie groups