Low-dimensional Lie Groups Research Articles

Universality in long-distance geometry and quantum complexity

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class1. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds2 by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application of these ideas of particular interest to physics and computer science is complexity geometry3–7—the study of quantum computational complexity using Riemannian geometry. We argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered. We conjecture that a new effective metric emerges at larger complexities that describes a broad class of complexity geometries, insensitive to various choices of microscopic penalty factors. We discuss the implications for recent conjectures in quantum gravity.

The Normalizer of a Lie Group: Applications and Challenges

Let G be a connected Lie group with Lie algebra g. This review is devoted to studying the fundamental dynamic properties of elements in the normalizer NG of G. Through an algebraic characterization of NG, we analyze the different dynamics inside the normalizer. NG contains the well-known left-invariant vector fields and the linear and affine vector fields on G. In any case, we show the shape of the solutions of these ordinary differential equations on G. We give examples in low-dimensional Lie groups. It is worth saying that these dynamics generate the linear and bilinear control systems on Euclidean spaces and the invariant and linear control systems on Lie groups. Moreover, the Jouan Equivalence Theorem shows how to extend this theory to control systems on manifolds.

Fermion Sampling: A Robust Quantum Computational Advantage Scheme Using Fermionic Linear Optics and Magic Input States

Fermionic Linear Optics (FLO) is a restricted model of quantum computation which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme which is a fermionic analogue of Boson Sampling: Fermion Sampling with magic input states. We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parity and is closely related to Matchgate circuits introduced by Valiant. Mathematically, these classes of circuits can be understood as fermionic representations of the Lie groups $U(d)$ and $SO(2d)$. This observation allows us to prove our main technical results. We first show anticoncentration for probabilities in random FLO circuits of both kind. Moreover, we prove robust average-case hardness of computation of probabilities. To achieve this, we adapt the worst-to-average-case reduction based on Cayley transform, introduced recently by Movassagh, to representations of low-dimensional Lie groups. Taken together, these findings provide hardness guarantees comparable to the paradigm of Random Circuit Sampling. Importantly, our scheme has also a potential for experimental realization. Both passive and active FLO circuits are relevant for quantum chemistry and many-body physics and have been already implemented in proof-of-principle experiments with superconducting qubit architectures. Preparation of the desired quantum input states can be obtained by a simple quantum circuit acting independently on disjoint blocks of four qubits and using 3 entangling gates per block. We also argue that due to the structured nature of FLO circuits, they can be efficiently certified.

Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi–Lie systems

Using the adjoint representations of Lie algebras, we classify all Jacobi structures on real two- and three-dimensional Lie groups. Also, we study Jacobi-Lie systems on these real low-dimensional Lie groups. Our results are illustrated through examples of Jacobi-Lie Hamiltonian systems on some real two- and three-dimensional Lie groups.

Classifying spaces for commutativity of low-dimensional Lie groups

AbstractFor each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras

The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensonal Lie group G is Liouville integrable. We derive this property from the fact that the coadjoint orbits of G are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension. We also give normal forms for left-invariant Riemannian and sub-Riemannian metrics on 3-dimensional Lie groups focusing on the case of solvable groups, as the cases of SO(3) and SL(2) have been already extensively studied. Our description is explicit and is given in global coordinates on G which allows one to easily obtain parametric equations of geodesics in quadratures.

Higgs bundles and exceptional isogenies

We explore relations between Higgs bundles that result from isogenies between low-dimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto $$\mathrm {SO}(4,\mathbb {C})$$ and $$\mathrm {SO}(6,\mathbb {C})$$ and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated with orthogonal Higgs bundles. In the case of $$\mathrm {SO}(6,\mathbb {C})$$ , our construction can be interpreted as a new description of Recillas’ trigonal construction.

Complex, Symplectic, and Contact Structures on Low-Dimensional Lie Groups

It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kahlerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.

Generalized harmonic functions and unitary representations of low dimensional Lie groups

The unitary representations of the three dimensional simple Lie groups are reconsidered from the perspective of harmonic functions acting on certain manifolds related to differential realisations of the groups themselves. By means of contractions of Lie groups, the procedure is also applied to the group E2 of rotations-translations in two dimensions.

Low dimensional Lie groups admitting left invariant flat projective or affine structures

We prove that any real Lie group of dimension ⩽5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension ⩽5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.

The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group G form a compact space C(G). We analyse the structure of C(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group H. We prove that C(H) is a 6-dimensional space that is path-connected but not locally connected. The lattices in H form a dense open subset L(H) ?C(H) that is the disjoint union of an infinite sequence of pairwise homeomorphic aspherical manifolds of dimension six, each a torus bundle over (S3 T) × R, where T denotes a trefoil knot. The complement of L(H) in C(H) is also described explicitly. The subspace of C(H) consisting of subgroups that contain the centre Z(H) is homeomorphic to the 4-sphere, and we prove that this is a weak retract of C(H).

Representations of Knowledge in Complex Systems

SUMMARY Modern sensor technologies, especially in biomedicine, produce increasingly detailed and informative image ensembles, many extremely complex. It will be argued that pattern theory can supply mathematical representations of subject-matter knowledge that can be used as a basis for algorithmic ‘understanding’ of such pictures. After a brief survey of the basic principles of pattern theory we shall illustrate them by an application to a concrete situation: high magnification (greater than 15000×) electron micrographs of cardiac muscle cells. The aim is to build algorithms for automatic hypothesis formation concerning the number, location, orientation and shape of mitochondria and membranes. For this we construct a pattern theoretic model in the form of a prior probability measure on the space of configurations describing these hypotheses. This measure is synthesized by solving sequentially a jump–diffusion equation of generalized Langevin form. The jumps occur for the creation–annihilation of hypotheses, corresponding to a jump from one continuum to another in configuration (hypothesis) space. These continua (subhypotheses) are expressed in terms of products of low dimensional Lie groups acting on the generators of a template. We use a modified Bayes approach to obtain the hypothesis formation, also organized by solving a generalized Langevin equation. to justify this it is shown that the resulting jump-diffusion process is ergodic so that the solution converges to the desired probability measure. to speed up the convergence we reduce the computation of the drift term in the stochastic differential equation analytically to a curvilinear integral, with the random term computed almost instantaneously. The algorithms thus obtained are implemented, both for mitochondria and membranes, on a 4000 processor parallel machine. Photographs of the graphics illustrate how automatic hypothesis formation is achieved. This approach is applied to deformable neuroanatomical atlases and tracking recognition from narrow band and high resolution sensor arrays.

Local Lie semigroups and open embeddings into global topological semigroups

The semigroup version of Lie's Third Fundamental Theorem asserts that each strictly positive cone and each finite-dimensional Lie wedge is a tangent wedge of some local semigroup. This paper investigates the question whether this local semigroup can be embedded into a global topological semigroup in such a fashion that the embedding preserves all existing products and its image is open. In the case of a strictly positive cone (in particular a finite-dimensional pointed cone) this question will be answered affirmatively. This result contrasts the situation of open embeddings into global subsemigroups of Lie groups, where counterexamples even in low-dimensional Lie groups occur. In the finite-dimensional case a homotopy-like congruence on the semigroup of conal curves induces a quotient semigroup (called conal path semigroup) which also solves the topological embedding problem.