Lie Subgroup Research Articles

Systems of first order ordinary differential equations allowing a given 3-dimensional Lie group as a subgroup of their symmetry group

AbstractWe determine systems of the first order ordinary differential equations such that their group of symmetries contains a three-dimensional Lie subgroup G. We represent the basis vectors of the Lie algebra $$\mathfrak {g}$$ g of G by vector fields in the three-dimensional real space. Two cases are distinguished according to whether the infinitesimal generators of $$\mathfrak {g}$$ g do not contain any component or contain component with respect to the independent variable of the system.

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

We study the relation between two special classes of Riemannian Lie groups [Formula: see text] with a left-invariant metric [Formula: see text]: The Einstein Lie groups, defined by the condition [Formula: see text], and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups [Formula: see text] are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the [Formula: see text]-invariant geodesic orbit metrics on Lie groups [Formula: see text] for a wide class of subgroups [Formula: see text] that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

Time optimal problems on Lie groups and applications to quantum control

<abstract><p>In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control (<sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $.</p> <p>In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by (<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.</p></abstract>

Dispersive problem and control sets of linear control systems on Lie groups

This paper is dedicated to the study of dispersiveness and controllability of linear control systems on Lie groups. Dispersiveness means absence of recursiveness, that is contrary to the existence of control set. A linear control system on a Lie group associates with a derivation operator. For a linear system with stable derivation, it is shown that the system is dispersive if and only if the trajectories through the neutral element have no limit at infinity. As a consequence, a nonempty limit set at the neutral element is a necessary condition for the linear system to admit a control set or to be controllable. In the nondispersive case, the control sets are described by the Lie subgroup of all recurrent points of the automorphism flow of the system. If the derivation operator is asymptotically stable, the central limit set at the neutral element is the unique control set of the system.

Motivic Atiyah-Segal Completion Theorem

Abstract Let $T$ be a torus, $X$ a smooth separated scheme of finite type equipped with a $T$-action, and $[X/T]$ the associated quotient stack. Given any localizing ${\mathbb {A}}^{1}$-homotopy invariant of dg categories $E$ (homotopy $K$-theory, algebraic $K$-theory with coefficients, étale $K$-theory with coefficients, $l$-adic algebraic $K$-theory, $l$-adic étale $K$-theory, semi-topological $K$-theory, topological $K$-theory, periodic cyclic homology, etc), we prove that the derived completion of $E([X/T])$ at the augmentation ideal $I$ of the representation ring $R(T)$ of $T$ agrees with the classical Borel construction associated to the $T$-action on $X$. Moreover, for certain localizing ${\mathbb {A}}^{1}$-homotopy invariants, we extend this result to the case of a linearly reductive group scheme $G$. As a first application, we obtain an alternative proof of Krishna’s completion theorem in algebraic $K$-theory, of Thomason’s completion theorem in étale $K$-theory with coefficients, and also of Atiyah-Segal’s completion theorem in topological $K$-theory (for those topological $M$-spaces $X^{\textrm {an}}$ arising from analytification; $M$ is a(ny) maximal compact Lie subgroup of $G^{\textrm {an}}$). These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason’s completion theorem the base field $k$ no longer needs to be separably closed, and in the case of Atiyah-Segal’s completion theorem the topological $M$-space $X^{\textrm {an}}$ no longer needs to be compact and the $M$-equivariant topological $K$-theory groups of $X^{\textrm {an}}$ no longer need to be finitely generated over the representation ring $R(M)$. As a second application, we obtain new completion theorems in $l$-adic étale $K$-theory, in semi-topological $K$-theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, $l$-adic, morphic, Betti, de Rham, etc) in terms of derived completion. Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy $K$-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of semi-topological $K$-theory.

The Fubini's Theorem for the Even Index Lie Subgroup

In this paper we present a new proof to generalize the Fubini's Theorem for the even index subgroup of a compact Lie group, using a recent result on the semi-direct product of groups. We also present some applications of this result for the invariant theory.

Geodesics as products of one‐parameter subgroups in compact lie groups and homogeneous spaces

AbstractWe study the geodesic equation for compact Lie groups G and homogeneous spaces , and we prove that the geodesics are orbits of products of one‐parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group , we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

The Heisenberg Lie Group is the most frequently used model for studying the representation theory of Lie groups. This Lie group is modular-noncompact and its Lie algebra is nilpotent. The elements of Heisenberg Lie group and algebra can be expressed in the form of matrices of size 3×3. Another specialty is also inherited by its three-dimensional Lie algebra and is called the Lie Heisenberg algebra. The Heisenberg Lie Group whose Lie Algebra is extended to the dimension 2n+1 is called the generalized Heisenberg Lie group and it is denoted by H whose Lie algebra is h_n. In this study, the surjectiveness of exponential mapping for H was studied with respect to h_n=⟨x ̅,y ̅,z ̅⟩ whose Lie bracket is given by [X_i,Y_i ]=Z. The purpose of this research is to prove the characterization of the Lie subgroup with respect to h_n. In this study, the results were obtained that if ⟨x ̅,y ̅ ⟩=:V⊆h_n a subspace and a set {e^(x_i ) e^(x_j ) ┤| x_i,x_j∈V }=:L⊆H then L=H and consequently Lie(L)≠V.

Generalized Lie symmetries and almost regular Lagrangians: a link between symmetry and dynamics

The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on the evolution of dynamical systems is determined. It is found that if the action has a generalized Lie symmetry, then the Lagrangian is necessarily singular; the converse is not true, as we show with a specific example. It is also found that the generalized Lie symmetry of the action is a Lie subgroup of the generalized Lie symmetry of the Euler–Lagrange equations of motion. The converse is once again not true, and there are systems for which the Euler–Lagrange equations of motion have a generalized Lie symmetry while the action does not, as we once again show through a specific example. Most importantly, it is shown that each generalized Lie symmetry of the action contributes one arbitrary function to the evolution of the dynamical system. The number of such symmetries gives a lower bound to the dimensionality of the family of curves emanating from any set of allowed initial data in the Lagrangian phase space. Moreover, if second- or higher-order Lagrangian constraints are introduced during the application of the Lagrangian constraint algorithm, these additional constraints could not have been due to the generalized Lie symmetry of the action.

Second-order sliding-mode on SO(3) and fault-tolerant spacecraft attitude control

As one of attitude representation methods for a rigid body, the rotation matrices defined on the special orthogonal matrix group SO(3) can give the unique, global and nonsingular solution. For the attitude control systems evolving on Lie group SO(3)×R3, a second-order sliding surface is constructed and proved to be a Lie subgroup, along which the reduced-order dynamics is almost globally asymptotically stable at the equilibrium state. Based on the new sliding-mode surface, a fault-tolerant controller is designed to deal with the actuator faults and external disturbances in attitude control systems. The new method is numerically simulated and further implemented in a semi-physical simulation system of an air-bearing platform to verify its effectiveness in spacecraft practical engineering.

FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory $${\varPhi }$$ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra $$H^{\text {cut}}_{\text {FG}}({\varPhi })$$ and lattices of intermediate structures. Feynman diagrams in $$H_{\text {FG}}({\varPhi })$$ are applied to describe states in $${\varPhi }$$ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space $$\mathcal {S}^{{\varPhi },M \subseteq [0,\infty )}_{\text {graphon}}([0,1])$$ . We associate Hopf subalgebras of $$H_{\text {FG}}({\varPhi })$$ generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green’s functions of $${\varPhi }$$ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.

High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics

In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge–Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space M is a manifold where a group G acts transitively. Such a space can be understood as a quotient M≅G/H, where a closed Lie subgroup H, is the isotropy group of each point of M. The Lie algebra of G may be decomposed into g=m⊕h, where h is the subalgebra that generates H and m is a subspace spanning the tangent spaces of M. Thus, variational problems on M can be treated as nonholonomically constrained problems on G, by requiring variations to remain on m. Nonholonomic partitioned RKMK integrators are derived as a modification of those obtained by a discrete variational principle on Lie groups, and can be interpreted as obeying a discrete Chetaev principle. These integrators tend to preserve several properties of their purely variational counterparts.

Lie theory for asymptotic symmetries in general relativity: The NU group

We study the Newman–Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi–Metzner–Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein.

Holomorphic Lie group actions on Danielewski surfaces

We prove that any Lie subgroup G (with finitely many connected components) of an infinite-dimensional topological group which is an amalgamated product of two closed subgroups can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).

Prolongations of G-structures related to Weil bundles and some applications

Let M be a smooth manifold of dimension m ≥ 1 and P be a G-structure on M , where G is a Lie subgroup of linear group GL(m). In [8], it has been defined the prolongations of G-structures related to tangent functor of higher order and some properties have been established. The aim of this paper is to generalize these prolongations to a Weil bundles. More precisely, we study the prolongations of G-structures on a manifold M , to its Weil bundle TAM (A is a Weil algebra) and we establish some properties. In particular, we characterize the canonical tensor fields induced by the A-prolongation of some classical G-structures.

Symmetric states and dynamics of three quantum bits

The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.

B-Structures on Lie groups and Poisson reduction

Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a b-Lie group as a pair (G,H) where G is a Lie group and H is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical b-symplectic structure on the b-cotangent bundle T⁎bG together with its reduction theory. Namely, we extend the minimal coupling procedure to T⁎bG/H and prove that the Poisson reduction under the cotangent lifted action of H by left translations can be described in terms of the Lie Poisson structure on h⁎ (where h is the Lie algebra of H) and the canonical b-symplectic structure on T⁎b(G/H), where G/H is viewed as a one-dimensional b-manifold having as critical hypersurface (in the sense of b-manifolds) the identity element.

Cooking pasta with Lie groups

We extend the (gauged) Skyrme model to the case in which the global isospin group (which usually is taken to be SU(N)) is a generic compact connected Lie group G. We analyze the corresponding field equations in (3+1) dimensions from a group theory point of view. Several solutions can be constructed analytically and are determined by the embeddings of three dimensional simple Lie groups into G, in a generic irreducible representation. These solutions represent the so-called nuclear pasta state configurations of nuclear matter at low energy. We employ the Dynkin explicit classification of all three dimensional Lie subgroups of exceptional Lie group to classify all such solutions in the case G is an exceptional simple Lie group, and give all ingredients to construct them explicitly. As an example, we construct the explicit solutions for G=G2. We then extend our ansatz to include the minimal coupling of the Skyrme field to a U(1) gauge field. We extend the definition of the topological charge to this case and then concentrate our attention to the electromagnetic case. After imposing a “free force condition” on the gauge field, the complete set of coupled field equations corresponding to the gauged Skyrme model minimally coupled to an Abelian gauge field is reduced to just one linear ODE keeping alive the topological charge. We discuss the cases in which such ODE belongs to the (Whittaker-)Hill and Mathieu types.

Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3)

<p style='text-indent:20px;'>This paper considers the attitude tracking control problem for a rigid body. In order to avoid the complexity and ambiguity associated with other attitude representations (such as Euler angles or quaternions), the attitude dynamics and the proposed control system are represented globally on special orthogonal groups. An adaptive controller based on a Lie subgroup of SO(3) is developed such that the rigid body can track any given attitude command asymptotically without requiring the exact knowledge of the inertia moment. In the presence of external disturbances, the adaptive controller is enhanced with an additional robust sliding mode term by following the same idea within the framework of SO(3). Finally, simulation results are presented to demonstrate efficiency of the proposed controllers.</p>

Seeking SUSY fixed points in the 4 \u2212 \u03f5 expansion

We use the 4 − ϵ expansion to search for fixed points corresponding to 2 + 1 dimensional mathcal{N} =1 Wess-Zumino models of NΦ scalar superfields interacting through a cubic superpotential. In the NΦ = 3 case we classify all SUSY fixed points that are perturbatively unitary. In the NΦ = 4 and NΦ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For NΦ = 4 we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For NΦ = 5, we go through all Lie subgroups of O(5) and use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with NΦ = frac{mathrm{N}left(mathrm{N}-1right)}{2} − 1 and O(N)/Z2 symmetry, that exists for arbitrary integer N≥3.