A quantum algebra approach to discrete equations on uniform lattices

A quantum algebra method for deducing the symmetries of discrete equations on uniform lattices is proposed. In principle, such a procedure can be applied to discretizations in a single coordinate (space or time) and the symmetries obtained in this way are indeed differential-difference operators. Firstly, the method is illustrated on two known examples that have been also analysed from the usual Lie symmetry approach: a uniform space lattice discretization of the (1+1) free heat-Schrödinger equation associated to a quantum Schrödinger algebra, and a discrete space (1+1) wave equation provided by a quantumso(2, 2) algebra. Furthermore, we construct a discrete space (2+1) wave equation from a new quantumso(3, 2) algebra, to show that this method is useful in higher dimensions. Time discretizations are also commented.     

Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.     

Contractions, Deformations and Curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-orthogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley–Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such “quantum” spaces.     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

Symmetry and general symmetry groups of the coupled Kadomtsev--Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev--Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac--Moody--Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et al。 From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.     

A Lie algebra structure on variation vector fields along curves in 2-dimensional space forms

A Lie algebra structure on variation vector fields along an immersed curve in a 2-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation are finally discussed from this point of view.     

Pre-symplectic algebroids and their applications

In this paper, we introduce the notion of a pre-symplectic algebroid and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated with a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi–Civita connection in the corresponding pseudo-Riemannian Lie algebroid.     

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.     

The reduction of the Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries

We prove a general theorem which allows the determination of Lie symmetries of the Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of the Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of the Laplace equation are directly related to the transition of the CKVs from the space where the original equation is defined to the space where the reduced equation resides. In particular we consider the reduction of the Laplace equation (i.e., the wave equation) in the Minkowski space and obtain the results of all previous studies in a straightforward manner. We consider the reduction of Laplace equation in spaces which admit Lie point symmetries generated from a non-gradient HV and a proper CKV and we show that the reduction with these vectors does not produce Type II hidden symmetries. We apply the results to general relativity and consider the reduction of Laplace equation in locally rotational symmetric space times (LRS) and in algebraically special vacuum solutions of Einstein’s equations which admit a homothetic algebra acting simply transitively. In each case we determine the Type II hidden symmetries.     

On the integrability of representations of finite dimensional real Lie algebras

We give an integrability criterion for Lie algebra representations in a reflexive Banach space. Applications are given to skewsymmetric Lie algebra representations in Hilbert spaces and to essential skewadjointness of a sum of two skewadjoint operators.     

Algebraical comparison of classical and quantum polynomial observables

The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure. The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not isomorphic, and hence there is no Dirac quantisation.     

A left-symmetric algebraic approach to left invariant flat (pseudo-)metrics on Lie groups

Left invariant flat metrics on Lie groups are revisited in terms of left-symmetric algebras which correspond to affine structures. There is a left-symmetric algebraic approach with an explicit formula to the classification theorem given by Milnor. When the positive definiteness of the metric is replaced by nondegeneracy, there are many more examples of left invariant flat pseudo-metrics, which play important roles in several fields in geometry and mathematical physics. We give certain explicit constructions of these structures. Their classification in low dimensions and some interesting examples in higher dimensions are also given.     

Compatible Lie Bialgebras

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket.We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra.They can also be regarded as a "compatible version" of Lie bialgebras,that is,a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra.Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie bialgebras are presented.In particular,there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in Lie algebras.Furthermore,a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter.As a byproduct,the compatible Lie bialgebras St into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.     

Physical vacuum properties and internal space dimension

The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization, however, is not unique. The case of 7-dimensional Riemannian space of signature 7(−) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of the Lie algebra E ₈. Considerations are presented, from which it follows that the least-dimen-si-on space bearing on physics is the Riemannian 11-dimensional space of signature 1(−)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density. Alexander Vasil'evich Pushkin was born on 13 April 1947 in St. Petersburg and died on 17 August 2004 in Sarov.     

Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1

We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev–Petviashvili (of type B) equation.     

Local Currents for a Deformed Heisenberg–Poincaré Lie Algebra of Quantum Mechanics,and Anyon Statistics

We set out to construct a Lie algebra of local currents whose space integrals, or “charges”, form a subalgebra of the deformed Heisenberg–Poincaré algebra of quantum mechanics discussed by Vilela Mendes, parameterized by a fundamental length scale ℓ. One possible technique is to localize with respect to an abstract single-particle configuration space having one dimension more than the original physical space. Then in the limit ℓ→0, the extra dimension becomes an unobservable, internal degree of freedom. The deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents modeled on (2+1)-dimensional space-time. This suggests a new possible interpretation of such representations of the local current algebra, not as describing conventional particles satisfying bosonic, fermionic, or anyonic statistics in two-space, but as describing systems obeying these statistics in a deformed one-dimensional quantum mechanics. In this context, we have an interesting comparison with earlier results of Hansson, Leinaas, and Myrheim on the dimensional reduction of anyon systems. Thus motivated, we introduce irreducible, anyonic representations of the deformed global symmetry algebra. We also compare with the technique of localizing currents with respect to the discrete positional spectrum.     

Lie 2-Bialgebras

In this paper, we study Lie 2-bialgebras, paying special attention to coboundary ones, with the help of the cohomology theory of L _∞-algebras with coefficients in L _∞-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras (also known as pre-Lie algebras) and symplectic Lie algebras (also called quasi-Frobenius Lie algebras).     

Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.     

Local symmetries of massless wave equations

The structure of a wave-equation symmetry algebra for massless particles is studied. It is shown that the local-symmetry algebra of massless wave equations belongs to the enveloping algebra of a conformal-group Lie algebra. It is also proven that the space of nontrivial symmetries of fixed order is irreducible under adjoint representation of the conformal algebra, and its dimension is indicated. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–26, May, 1998.