Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L₀ that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L₀ has finite codimension in L and there are Lie subalgebras L₀=L⁰⊂L¹⊂?⊂Lp=L such that Li+1=Li+[Li,Li+1] for all i; (2) L₀ is a Lie ideal of L and dim(L₀)=∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.     

Strongly Flat and PO-Flat S-Posets

Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.     

Group analysis of the von Kármán–Howarth equation. Part I. Submodels

We study the von Kármán–Howarth (KH) equation by group theoretical methods. This scalar partial differential equation involves two dependent variables (closure problem) and, it has been derived from the Navier–Stokes equations. The equivalence Lie algebra L has been found to be infinite-dimensional and, it is spanned by the four operators. The subalgebra of L is spanned by the three operators. Furthermore, ideal comprises one operator. Optimal systems of one-, two- and three-dimensional subalgebras have been obtained. Normalizers for the one- and two-dimensional subalgebras have been calculated. Finally we have obtained the submodels of the KH equation corresponding to optimal system of one- and two-dimensional subalgebras. This merely suggests alternative solutions to the closure problem of isotropic turbulence.     

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g^∗), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g^∗. We write down families of examples for the case that g is a classical complex simple Lie algebra.     

Some Theorems on Leibniz Algebras

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.     

Similarity variables and reduction of the heat equation on torus

The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £₁ and £₂ the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £₃ and £₄ the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution.     

Les sous-algebres de Cartan reelles et la frontiere d'une orbite ouverte dans une variete de drapeaux

We show how the non compact imaginary roots of a non compact real semi-simple Lie algebra with respect to a Cartan subalgebra to allows us, alike the real roots of, to give a complete classification of the G-conjugacy classes of Cartan subalgebras of if G_c is a complex connected group whose algebra is the complexified of, if B is a Borel subgroup of G_c and G the analytic subgroup of G_c corresponding to the subalgebra of, we determine the G-orbits of codimension one in the boundary of an open G-orbit of the complex flag manifold G_c/B. If is a maximally compact Cartan subalgebra of contained in, we show how the imaginary non compact simple roots of allows us to determine such orbits.     

Cohomology of the Lie algebra ℌ2: Experimental results and conjectures

The cohomology with trivial coefficients of the Lie algebra ? of Hamiltonian vector fields in the plane and of its maximal nilpotent subalgebra L ₁? is considered. The cohomology H ₂(L ₁?) is calculated, and some far-reaching conjectures concerning the cohomology of the Lie algebras mentioned above and based on an extensive experimental material are formulated.     

Classification of Differential Invariant Submodels

Calculation of differential invariants and invariant differentiation operators of a subalgebra of the Lie algebra admitted by a system of differential equations enables us to construct differential invariant submodels. We classify submodels for every subalgebra of an optimal system of subalgebras. Classification includes the invariant submodels and partially invariant submodels considered earlier. We give examples of classification for three-dimensional subalgebras admitted by the equations of gas dynamics.     

Almost Primitive Elements of Free lie Algebras of Small Ranks

Let K be a field, X = {x1, . . . , xn}, and let L(X) be the free Lie algebra over K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset M of nonzero elements of the free Lie algebra L(X) is said to be primitive if there is a set Y of free generators of L(X), L(X) = L(Y ), such that M ? Y (in this case we have |Y | = |X| = n). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element u of the free Lie algebra L(X) is said to be almost primitive if u is not a primitive element of the algebra L(X), but u is a primitive element of any proper subalgebra of L(X) that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.     

Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus g = 1, 2,.. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators L _2q , q = ?1, 0, 1, 2,.., of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.     

Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A₁, A₁⊕A₂, 3A₁, or A3,3⊕A₁. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A₁ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250].     

Special Lie superalgebras with maximality condition for subalgebras

We consider finitely generated Lie superalgebras over a field of characteristic zero satisfying Capelli identities. We prove that any such an algebra with the maximality condition for abelian subalgebras is finite dimensional. In particular, any special Lie superalgebra with the maximality condition for its subalgebras has a finite dimension. We also prove that the universal enveloping algebra U(L) of special Lie superalgebra L is Noetherian if and only if $\dim L<\infty$ .     

Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.     

Symmetry analysis of a system of modified shallow-water equations

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun Nonlinear Sci Numer Simulat 2012;20:630–36]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are found from solutions of the linearized MSWE.     

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < . We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.     

The Ideal Index of Subalgebras of a Finite Dimensional Lie Algebra

D. A. Towers introduced the notion of ideal index of a maximal subalgebra of a Lie algebra, and used it to analyze the influence of maximal subalgebras on the structure of a finite dimensional Lie algebras.

In this article, we generalize the ideal index from maximal subalgebras to all subalgebras, and obtain some new characterizations of solvable and supersolvable Lie algebras by the ideal indices of some certain subalgebras.