Three-dimensional Real Lie Algebras Research Articles

Levi flat CR structures on 3D Lie algebras

We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties.

Classification of three-dimensional zeropotent algebras over the real number field

ABSTRACTA nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.

Classical r-matrices via semidualisation

We study the interplay between double cross sum decompositions of a given Lie algebra and classical r-matrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an expression for classical r-matrices of the semidual Lie bialgebra in terms of the data which determines the decomposition of the original Lie algebra. Applied to the local isometry Lie algebras arising in three-dimensional gravity, decomposition, and semidualisation yields the main class of non-trivial r-matrices for the Euclidean and Poincaré group in three dimensions. In addition, the construction links the r-matrices with the Bianchi classification of three-dimensional real Lie algebras.

Jacobi operators of quantum counterparts of three-dimensional real Lie algebras over the harmonic oscillator

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional real Lie algebras. The Jacobi operators of these quantum algebras are explicitly calculated.

Quantum counterparts of three-dimensional real Lie algebras over harmonic oscillator

Abstract Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional (3D) real Lie algebras in Bianchi classification. The Jacobi operators of the quantum algebras are found.

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.

The three-dimensional real Lie algebras and their contractions

In this paper, the general contractions of the three-dimensional real Lie algebras are determined. A short summary will be given of the history, definitions, properties, examples, and applications of contractions of Lie algebras. So far, mostly simple or generalized Inönü–Wigner contractions have been used. The three-dimensional real Lie algebras are classified in such a way that their general contractions can be easily read off. All these contractions can be realized by generalized Inönü–Wigner contractions, which supports the argument that these are the interesting objects to study.

Geometry of crystal structure with defects. II. Non-Euclidean picture

The following point of view is geometrically formulated and its consequences examined: the lattice of a crystalline body with a continuous distribution of dislocations can be locally described as an ideal lattice in non-Euclidean space. The types of distribution of dislocations are described by the classification of three-dimensional real Lie algebras. The influence of point defects and the elastic deformation field on the geometry of the material structure of a crystalline body with dislocations is examined. The case where a crystal with dislocations reacts as a body with internal rotational degrees of freedom is discussed.