Search Results (16)

The present paper elaborates on the development of a theory of discrete-time dynamical systems on finite-dimensional structured state spaces. Dynamical systems on structured state spaces possess well-known applications to solving differential equations in physics, and it was shown that discrete-time systems on finite- (albeit high-) dimensional structured state spaces possess solid applications to structured signal processing and nonlinear system identification, modeling and control. With reference to the state-space representation of dynamical systems, the present contribution tackles the core system-theoretic problem of determining suitable laws to express a system’s state transition. In particular, the present contribution aims at formulating a fairly general class of state-transition laws over the Lie algebra associated to a Lie group and at extending some properties of classical dynamical systems to process Lie-algebra-valued state signals. Full article

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional $N=1,2,3$- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable $N=2,3$-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra $\mathcal{K}(1|3)$ and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the $N=1,2,3$-supercircle. Full article

The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{a_i\right\}$ with n punctures $a_i$ are encoded in its fundamental group ${\pi }_1\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_1$ over a Lie group G as globalized by the character variety $\mathcal{C}=\mathrm{Hom}({\pi }_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere $\mathsf{\Sigma }=S_2^{\left(4\right)}$ and the ‘space-time-spin’ group $G=S{L}_2\left(\mathbb{C}\right)$. In such a situation, $\mathcal{C}$ possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or $P_{VI}$); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of $P_{VI}$. In this paper, we feature the parametric representation of some solutions of $P_{VI}$: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups $f_p$ encountered in TQC or DNA/RNA sequences are proposed. Full article

This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy–momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy–momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress–energy. Full article

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions $H_s\left(z\right)$ comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters $p_s>0$, $s=1,\dots ,n$. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks $n=1,2,3,4,5$ and exceptional algebra $E_6$. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored. Full article

GIM Lie algebras are the generalizations of Kac–Moody Lie algebras. However, the structures of GIM Lie algebras are more complex than the latter, so they have encountered many new difficulties to study their representation theory. In this paper, we classify all finite dimensional simple modules over the GIM Lie algebra ${\mathcal{Q}}_{n+1}(2,1)$ as well as those over ${\Theta }_{2n+1}$. Full article

In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the system to 12 and 14 parameters, respectively. Invariant submodels of rank one are constructed from two three-dimensional subalgebras of the corresponding Lie algebras, and exact solutions with a linear velocity field with inhomogeneous deformation are obtained. On the one hand of the special state equation, the submodel describes an isochoric vortex motion of particles, isobaric along each world line and restricted by a moving plane. The motions of particles occur along parabolas and along rays in parallel planes. The spherical volume of particles turns into an ellipsoid at finite moments of time, and as time tends to infinity, the particles end up on an infinite strip of finite width. On the other hand of the state equation of a monatomic gas, the submodel describes vortex compaction to the origin and the subsequent expansion of gas particles in half-spaces. The motion of any allocated volume of gas retains a spherical shape. It is shown that for any positive moment of time, it is possible to choose the radius of a spherical volume such that the characteristic conoid beginning from its center never reaches particles outside this volume. As a result of the generalization of the solutions with a linear velocity field, exact solutions of a wider class are obtained without conditions of invariance of density and pressure with respect to the selected three-dimensional subalgebras. Full article

The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits. Full article

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to ${\mathbb{R}}^I\times C$ for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra $\mathbb{R}\left[G\right]$ of a finite group G to its representation algebra $R(G,\mathbb{R})$, via the natural duality of the topological vector space ${\mathbb{R}}^I$ to the vector space ${\mathbb{R}}^{\left(I\right)}$, for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups. Full article

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups $H_2$, $H_3$ and $H_4$. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups $H_2$ and $H_3$. The geometrical structures of nested polytopes are exemplified. Full article

We derive generalised uncertainty relations (GURs) for orbital angular momentum and spin in the recently proposed smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum and recovers both the generalised uncertainty principle (GUP) and extended uncertainty principle (EUP), previously proposed in the quantum gravity literature, within a single formalism. In this paper, we investigate the consequences of these results for particles with extrinsic and intrinsic angular momentum and obtain generalisations of the canonical $\mathrm{so}(3)$ and $\mathrm{su}(2)$ algebras. We find that, although $\mathrm{SO}(3)$ symmetry is preserved on three-dimensional slices of an enlarged phase space, corresponding to a superposition of background geometries, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for orbital angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, $ħ\to ħ+\beta$ . The value of the new parameter, $\beta \simeq ħ\times {10}^{-61}$ , is determined by the ratio of the dark energy density to the Planck density, and its existence is required by the presence of both minimum length and momentum uncertainties. Here, we assume the former to be of the order of the Planck length and the latter to be of the order of the de Sitter momentum $\sim ħ\sqrt{\mathsf{\Lambda }}$ , where $\mathsf{\Lambda }$ is the cosmological constant, which is consistent with the existence of a finite cosmological horizon. In the smeared-space model, ħ and $\beta$ are interpreted as the quantisation scales for matter and geometry, respectively, and a quantum state vector is associated with the spatial background. We show that this also gives rise to a rescaled Lie algebra for generalised spin operators, together with associated subalgebras that are analogous to those for orbital angular momentum. Remarkably, consistency of the algebraic structure requires the quantum state associated with a flat background to be fermionic, with spin eigenvalues $\pm \beta /2$ . Finally, the modified spin algebra leads to GURs for spin measurements. The potential implications of these results for cosmology and high-energy physics, and for the description of spin and angular momentum in relativistic theories of quantum gravity, including dark energy, are briefly discussed. Full article

In non-relativistic quantum mechanics, the dynamics of closed quantum systems is described by Hamiltonians which are self-adjoint in appropriately chosen Hilbert spaces. For PT-symmetric quantum systems, the Hamiltonians are, in general, no longer self-adjoint in standard Hilbert spaces, rather they are self-adjoint in Krein spaces—Hilbert spaces endowed with indefinite metric structures. Moreover, the spectra of PT-symmetric Hamiltonians are symmetric with regard to the real axis in the spectral plane. Apart from Hamiltonians with purely real spectra, this includes also Hamiltonians whose spectra may contain sectors of pairwise complex-conjugate eigenvalues. Considering families of parameter-dependent Hamiltonians, one can arrange parameter-induced passages from sectors of purely real spectra to sectors of complex-conjugate spectral branches. Corresponding passages can be regarded as PT-phase transitions from sectors of exact PT-symmetry to sectors of spontaneously broken PT-symmetry. Approaching a PT-phase transition point, the eigenvectors of the Hamiltonian tend toward their isotropic limit—an, in general, infinite-dimensional (Krein-space) generalization of the light-cone limit in Minkowski space. At a phase transition, the Hamiltonian is no longer diagonalizable, but similar to an arrangement of nontrivial Jordan-blocks. The interplay of these structures is briefly reviewed with special emphasis on the related Lie-algebraic and Lie-group aspects. With the help of Cartan-decompositions, associated hyperbolic structures and Lie-triple-systems are discussed for finite-dimensional setups as well as for their infinite-dimensional generalizations (Hilbert-Schmidt (HS) Lie groups, HS Lie algebras, HS Grassmannians). The interconnection of Krein-space structures and PT-phase transitions is demonstrated on two exactly solvable models: PT-symmetric Bose-Hubbard models and PT-symmetric plaquette arrangements. Full article

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M₀ x M₁ x . . . x M_n , where all M_i with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M₀ is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M₀ . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M₀ of dimension d₀ ≠ 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d₀ = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H₂(q, q) , AE₃, HA⁽¹⁾₂, E₁₀ and Lorentzian KM algebra P₁₀ , are presented. Full article

A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found. Full article

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups. Full article