Three-dimensional Lie Algebra Research Articles

BIANCHI IX GROUP-MANIFOLD REDUCTIONS OF GRAVITY

We exhibit a new way to perform the group-manifold reduction of pure Einstein gravity in the vielbein formulation when the compactification group manifold is S3. The new Bianchi IX group-manifold reduction is obtained by exploiting the two three-dimensional Lie algebras that the S3 group manifold admits. As an application of the new reduction we show that there exists a domain wall solution to the lower-dimensional theory which upon uplifting to the higher-dimension turns out to be the self-dual (in the nonvanishing components of both curvature and spin connection) Kaluza–Klein monopole.

Generalized Bessel Functions and Lie Algebra Representation

The generalized Bessel functions (GBF) are framed within the context of the representation Q(ω,m0) of the three-dimensional Lie algebra \(\mathcal{T}_{3}\) . The analysis has been carried out by generalizing the formalism relevant to Bessel functions. New generating relations and identities involving various forms of GBF are obtained. Certain known results are also mentioned as special cases.

Dynamical symmetries of semi-linear Schrödinger and diffusion equations

Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra ( conf 3 ) C . We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of ( conf 3 ) C are classified and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ALGEBRAS

We consider versal deformations of 0|3-dimensional L∞algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

Nice parabolic subalgebras of reductive Lie algebras

This paper gives a classification of parabolic subalgebras of simple Lie algebras over C \mathbb {C} that are complexifications of parabolic subalgebras of real forms for which Lynch’s vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three-dimensional simple Lie algebra).

Simultaneous Quantization of Three-Dimensional Bialgebras

We perform a simultaneous multiparameter quantization of some three-dimensional Lie algebras, such as the Heisenberg algebra, motion algebras of the Euclidean and pseudoeuclidean planes, and the algebra su(2). Such a quantization is possible since any of the mentioned algebras is dual to the same solvable Lie algebra. We present an explicit form of the numerical R-matrix; this allows us to represent some of the commutation relations in the form of the PTT-equation. Bibliography: 14 titles.

On noncommutative Noetherian schemes

The main aim of this paper is to better understand the localization technique for certain Noetherian rings like enveloping algebras of nilpotent Lie algebras. For such rings R we also give a conjectural definition of certain sheaves which should be “affine” objects naturally generalizing the classically defined structure sheaves in commutative theory. The corresponding sheaves associated to some R-modules might carry particularly interesting information; e.g., for representation theory of semisimple Lie groups. Next, we generalize one important theorem of P.F. Smith on localization in Noetherian Artin–Rees rings. As an interesting corollary we obtain that every prime ideal of height 1 in the enveloping algebra of the Lie algebra sl ( 2 ) over a field of characteristic zero is localizable. Finally, we provide a number of concrete useful calculations for our main example, the enveloping algebra of the three-dimensional Heisenberg Lie algebra; and thus test both the proposed ideas and methods. In particular, we introduce the notion of a weakly normal element, generalizing the notion of a normal element.

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable*1

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L . By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L ), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x ∈ L , x ∉Solv( L ), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p , p ), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L. By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x∈ L, x∉Solv( L), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p, p), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.

Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation

There are seven equivalence classes of second-order ordinary differential equations possessing only three Lie point symmetries and hence not linearisible by means of a point transformation. We examine the representatives of these classes for linearisibility by means of other types of transformation. In particular we compare the potential for linearisibility and the possession of the Painlevé property. The complete symmetry group is realised in the standard algebra for each of the equivalence classes.

Lie-theoretic generating relations of two variable Laguerre polynomials

Generating relations involving two variable laguerre polynomials L n ( x, y) are derived. The process involves the construction of a three-dimensional Lie algebra isomorphic to special linear algebra sl(2) with the help of Weisner's method by giving suitable interpretations to the index n of the polynomials L n ( x, y).

Self-Balance Equations and Bianchi-Type Distortions in the Theory of Dislocations

We deals with two aspects of the geometric description of continuized Bravais monocrystals with many dislocations. First, a triad of vector fields is distinguished constituting a basis for the C∞-module of smooth vector fields tangent to the body. This moving frame defines its object of anholonomity as well as an intrinsic (“material”) Riemannian metric of the body. Second, these geometric objects are used to define both the notions of principal congruence of Volterra-type effective dislocations and principal local Burgers vector associated with these congruences. The main topics discussed are (i) self-balance equations of dislocations and secondary point defects generated by distributions of these dislocations; (ii) a linkage of the Bianchi classification of three-dimensional real Lie algebras with the physical classification of principal local Burgers vectors.

Two-step solvable Lie algebras and weight graphs

In this paper the authors propose a new approach to the study of weight systems. Instead of considering graphs whose vertices correspond to the generators of a Lie algebra (as for Cartan subalgebras in the semisimple case), the authors consider the whole weight system. The purpose is to extract information about the weight system from the geometry of the weights. The considerations are restricted to the case where a torus of derivations induces a decomposition of a nilpotent Lie algebra g into one-dimensional weight spaces, none of which is associated with the zero weight. The paper is structured as follows: In Section 2 the most important facts of weight systems of nilpotent Lie algebras and the root system associated to solvable Lie algebras are recalled. In Section 3 the authors formulate their conditions on the weight systems and analyze the consequences of these conditions on the structure of the weight system. They also define associated weight graphs and deduce their elementary geometrical properties. This provides a characterization of the three-dimensional Heisenberg Lie algebra in terms of trees. Section 4 is devoted to the study of certain subgraphs of a weight graph which can be used to reconstruct the weight system from the weight graph. If r is a semidirect product of g and a torus T these subgraphs determine bounds for the solvability class of r . In Section 5 these results are applied to obtain a geometrical proof of the nonexistence of two-step solvable rigid Lie algebras.

On the Property M Conjecture for the Heisenberg Lie Algebra

We prove a fundamental case of a conjecture of the first author which expresses the homology of the extension of the Heisenberg Lie algebra by C [ t]/( t k+1 ) in terms of the homology of the Heisenberg Lie algebra itself. More specifically, we show that both the 0th and ( k+1)th x-graded components of homology of this extension of the three-dimensional Heisenberg Lie algebra have dimension 3 k+1 by constructing a simple basis for cohomology.

The Forms and Representations of the Lie algebra sl 2 (ℤ)

We study the structure of integral p-adic forms of the splitting three-dimensional simple Lie algebra over the field of p-adic numbers. We discuss the questions of diagonalizability of such forms and description for maximal diagonal ideals. We consider torsion-free finite-dimensional modules over the splitting three-dimensional simple Lie algebra with integral and p-adic integral coefficients. We describe diagonal modules, demonstrate finiteness of the number of modules in each dimension, and prove a local-global principle for irreducible modules.

AUTOMORPHISMS OF THE ENVELOPING ALGEBRA OF THE THREE-DIMENSIONAL NILPOTENT LIE ALGEBRA

We prove that the group of tame automorphisms of the universal enveloping algebra of the three-dimensional nilpotent nonabelian Lie algebra over a field of characteristic 0 is dense with respect to the formal power series topology in the group of all automorphisms. *Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research. E-mail: drensky@math.bas.bg

Lie algebras satisfying identities of degree 5

Let Φbe an associative ring with unity, containing 1/6.We prove that every prime Lie Φ-algebra satisfying the identity [(yx)(zx)]x = 0is embedded as a subring of a special form in a three-dimensional simple Lie algebra over some field A. It follows that there exists no central simple Lie algebra which is not three-dimensional and the cube of every inner derivation in which is a derivation. It is proved that if a semiprime Lie algebra over a field Φsatisfies an arbitrary identity of degree 5 (not following from the anticommutativity and Jacobi identities), then it also satisfies the standard identity of degree 5. Essentially used in the proof is the notion of antiderivation. In passing we show that every prime Lie algebra having a nonzero antiderivation satisfies the standard identity of degree 5.

New Treatment on Bifurcations of Periodic Solutions and Homoclinic Orbits at High r in the Lorenz Equations

In this paper, the existence of periodic solutions and homoclinic orbits in the Lorenz equations with high r is rigorously proved. The paper deals with the Lorenz model as a three-dimensional perturbed Hamiltonian system generated by the three-dimensional Lie algebra. By using the method of Melnikov vector,the explicit parametric conditions can be determined.

Some forms of Kac-Moody algebras

Let k be an algebraically closed field of char 0. The (unique up to isomorphism) three-dimensional simple Lie algebra s1(2, k) plays an important role in the theory of the semisimple Lie algebras over k. For example, Chevalley, Harish Chandra, and Serre’s construction of the semisimple Lie algebras g over k corresponding to a Dynkin diagram 9 can be viewed as “glueing together” copies of sl(2, k) in “Lf but we can not choose the distinct TDS in an arbitrary way. We need some compatibility conditions between the TDS attached to neighbouring vertices: they are described by some scalars sij. And for simplicity, we restrict our attention in this definition to Dynkin diagrams without cycles. For both definitions, we also make the additional hypothesis that the

Some remarkable combinatorial matrices

In this paper we describe, in combinatorial terms, some matrices which arise as Laplacians connected to the three-dimensional Heisenberg Lie algebra. We pose the problem of finding the eigenvalues and eigenvectors of these matrices. We state a number of conjectures including the conjecture that all eigenvalues of these matrices are non-negative integers. We determine the eigenvalues and eigenvectors explicitly for an important subclass of these matrices.