On a family of analytic diassociative loops

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media — cosmology, crystallography, etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin, etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and this paper is the first step in addressing this wide and diverse class of groups and algebras. This paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.

Classification of a family of 4-dimensional anti-commutative algebras and their automorphisms

After giving some definitions and results on Lie groups, Lie algebras and homogeneous spaces, this chapter discusses problems on these topics. First, some specific examples of Lie groups and Lie algebras are introduced. Then, we consider homomorphisms, Lie subgroups and Lie subalgebras, integration on Lie groups, the exponential map exp and its differential map exp∗, the adjoint representation Ad and its differential map ad, and Lie groups of transformations. We include several problems with the idea of introducing the reader to basic topics on Lie groups and Lie algebras, such as the determination of all the 2-dimensional Lie algebras, and all (up to isomorphism) 1-dimensional Lie groups. We also present some calculations that justify the name ‘exponential,’ the computation of the dimensions of some classical groups, and some properties of the Heisenberg group are studied. Some miscellaneous topics are considered as well. Special mention should also be paid of the fact that, in the present edition, two new problems have been added in the section concerning the exponential map, where the simply connected Lie group corresponding to a given Lie algebra is obtained, and that the section devoted to the adjoint representation, contains six new problems concerning somewhat more specialized topics such as Weyl group, Cartan matrix, Dynkin diagrams, etc. Similarly, the section devoted to Lie groups of transformations, has been increased in ten new application problems in Symplectic Geometry, Hamiltonian Mechanics, and other related topics. We then switch to homogeneous spaces, bringing in various classical examples, as the sphere, the real Stiefel manifold, and the real Grassmannian and other not so usual examples.

The condition of self-consistency for the quantum description of dissipative systems makes it necessary to exceed the limits of Lie algebras and groups, i.e., this requires the application of non-Lie algebras (in which the Jacobi identity does not hold) and analytic quasi-groups (which are nonassociative generalizations of groups). In the present paper, we show that the analogues of Lie algebras and groups for quantum dissipative systems are commutant-Lie algebras (anticommutative algebras whose communtants are Lie subalgebras) and analytic commutant-associative loops (whose commutants are associative subloops (groups)). It is proved that the tangent algebra of an analytic commutant-associative loop with invertibility (a Valya loop) is a commutant-Lie algebra (a Valya algebra). Some examples of commutant-Lie algebras are considered.

In this work, n-th nilpotent and minimal nilpotent products of parafree Lie algebras are examined and it is shown that these products are parafree. Also, a base set is obtained for the n-th nilpotent products of two abelian Lie algebras that are nilpotent of class n.In this work, n-th nilpotent and minimal nilpotent products of parafree Lie algebras are examined and it is shown that these products are parafree. Also, a base set is obtained for the n-th nilpotent products of two abelian Lie algebras that are nilpotent of class n. In this work, n-th nilpotent and minimal nilpotent products of parafree Lie algebras are examined and it is shown that these products are parafree. Also, a base set is obtained for the n-th nilpotent products of two abelian Lie algebras that are nilpotent of class n.

We study the problem of matrix isomorphism of matrix Lie algebras (MatIsoLie). Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley -- Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices that is closed under linear combinations and the operation [A, B] = AB - BA. Two matrix Lie algebras L, L' are matrix isomorphic if there is an invertible matrix M such that conjugating every matrix in L by M yields the set L'. We show that certain cases of MatIsoLie -- for the wide and widely studied classes of semi simple and abelian Lie algebras -- are equivalent to graph isomorphism and linear code equivalence, respectively. On the other hand, we give polynomial-time algorithms for other cases of MatIsoLie, which allow us to mostly derandomize a recent result of Kayal on affine equivalence of polynomials.

We study the problem of matrix isomorphism of matrix Lie algebras (MatIsoLie). Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley -- Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices that is closed under linear combinations and the operation [A, B] = AB - BA. Two matrix Lie algebras L, L' are matrix isomorphic if there is an invertible matrix M such that conjugating every matrix in L by M yields the set L'. We show that certain cases of MatIsoLie -- for the wide and widely studied classes of semi simple and abelian Lie algebras -- are equivalent to graph isomorphism and linear code equivalence, respectively. On the other hand, we give polynomial-time algorithms for other cases of MatIsoLie, which allow us to mostly derandomize a recent result of Kayal on affine equivalence of polynomials.

The class of binary Lie algebras contains the one of Malcev algebras (and so the one of Lie algebras). We provide a classification of central extensions of complex non-Malcev binary Lie algebras of dimensions either 3 or 4. From here, a classification of central extensions of complex non-Lie anticommutative ℭ𝔇-algebras of dimensions either 3 or 4 is also given.

We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for [Formula: see text]-step nilpotent Lie algebras determined by graphs, free [Formula: see text] and [Formula: see text]-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand, we also exhibit families of nilpotent Lie algebras having an arbitrary large space of non-inner almost inner derivations.

Balanced Hermitian structures on almost abelian Lie algebras

The inventors of Lie groups and Lie algebras (starting with Lie!) regarded Lie groups as groups of symmetries of various topological or geometric objects. Lie algebras were viewed as the “infinitesimal transformations” associated with the symmetries in the Lie group. For example, the group SO(n) of rotations is the group of orientation-preserving isometries of the Euclidean space \( \mathbb{E}^n \) . The Lie algebra so (n,ℝ) consisting of real skew symmetric n × n matrices is the corresponding set of infinitesimal rotations. The geometric link between a Lie group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space to the Lie group at the identity. There is a map from the tangent space to the Lie group, called the exponential map. The Lie algebra can be considered as a linearization of the Lie group (near the identity element), and the exponential map provides the “delinearization,” i.e., it takes us back to the Lie algebra. These concepts have a concrete realization in the case of groups of matrices, and for this reason we begin by studying the behavior of the exponential maps on matrices.

This paper is to classify Lorentzian Lie groups ( G , ⟨ · , · ⟩ ) $(G,\langle \cdot ,\cdot \rangle )$ admitting non-Killing left-invariant conformal vector fields whose Lie algebra is g ${\mathfrak {g}}$ . As we know, g ${\mathfrak {g}}$ is solvable and [ g , g ] $[{\mathfrak {g}},{\mathfrak {g}}]$ is of codimension 1 in g ${\mathfrak {g}}$ . In this paper, we will prove that [ g , g ] $[{\mathfrak {g}},{\mathfrak {g}}]$ is an Abelian Lie algebra, or a direct sum of a generalized Heisenberg Lie algebra and an Abelian Lie algebra. We obtain a simple criterion for such Lorentzian Lie groups with dim G ⩾ 4 $\dim G\geqslant 4$ to be conformally flat, and moreover, examples of conformally flat and non-conformally flat Lorentzian Lie groups are constructed. Finally, we prove that Lorentzian Lie groups admitting non-Killing left-invariant conformal vector fields can be shrinking, steady and expanding Ricci solitons.

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media—cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures, such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified and studied in details. However, a systematic investigation of almost Abelian Lie groups has not been carried out yet, and the present paper is devoted to an explicit description of properties of this wide and diverse class of groups. The subject of investigation are real almost Abelian Lie groups with their Lie group theoretical aspects, such as the exponential map, faithful matrix representations, discrete and connected subgroups, quotients and automorphisms. The emphasis is put on explicit description of all technical details.

The aim of this work is to completely describe two families of Lie algebras whose Lie groups have negative sectional curvature. The first family consists of Lie algebras satisfying the following property: given any two vectors in the Lie algebra, the linear subspace spanned by them is a Lie subalgebra. On the other hand, the second family consists of reduced Lie algebras of Iwasawa type.