Dimensional Lie Algebra Research Articles

On a Microscopic Representation of Space-Time III

Using the Dirac (Clifford) algebra $$\gamma ^{\mu }$$ as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15 dimensional Lie algebra su*(4) as complex embedding of sl(2, $$\mathbb {H}$$ ), the relation to the compact group SU(4) as well as subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of $$\mathbb {R}^{3}$$ . This line geometrical description, however, leads to applications and identifications of line Complexe and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac’s ‘square root of $$p^{2}$$ ’, the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook.

Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras

The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras.

Derivation algebra of direct sum of lie algebras

Let L1 and L2 be two finite dimensional Lie algebras on arbitrary field F with no common direct factor and L=L1⊕L2. In this article, we express the structure and dimension of derivation algebra of L, Der(L), and some of their subalgebras in terms of Der(L1), Der(L2), Hom(L1,Z(L2)), and Hom(L2,Z(L1)).

On the Lie symmetry analysis and traveling wave solutions of time fractional fifth-order modified Sawada-Kotera equation

In this paper, we study Lie symmetry analysis of the time fractional fifth-order modified Sawada-Kotera equation (FMSK) with Riemann-Liouville derivative. Applying the adapted the Lie group theory to the equation under study, two dimensional Lie algebra is deduced. Using the obtained nontrivial Lie point symmetry, it is shown that the equation can be converted into a nonlinear fifth order ordinary differential equation of fractional order in the meaning of the Erdelyi-Kober fractional derivative operator. In addition, we construct some exact traveling solutions for the FMSK using the sub-equation method.

On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank

We have recently shown that pseudo-bosonic operators realize concrete examples of finite dimensional nilpotent Lie algebras over the complex field. It has been the first time that such operators were analyzed in terms of nilpotent Lie algebras (under prescribed conditions of physical character). On the other hand, the general classification of a finite dimensional nilpotent Lie algebra l may be given via the size of its Schur multiplier involving the so-called corank t(l) of l. We represent l by pseudo-bosonic ladder operators for t(l)≤6 and this allows us to represent l when its dimension is ≤5.

Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili (KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra W1+∞. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which contain many interesting noncommutative integrable systems such as the noncommutative KdV hierarchy and noncommutative Boussinesq hierarchy. Meanwhile, we construct two new noncommutative systems including odd noncommutative C type Gelfand-Dickey and even noncommutative C type Gelfand-Dickey hierarchies. Also using the symmetry, we can construct a new noncommutative Gelfand-Dickey hierarchy with self-consistent sources. Based on the natural differential Lax operator of the noncommutative Gelfand-Dickey hierarchy, the string equations of the noncommutative Gelfand-Dickey hierarchy are also derived.

Dual Lie Bialgebra Structures of Twisted Schrödinger-Virasoro Type

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained.

The Grassmann Algebra and its Differential Identities

Let G be the infinite dimensional Grassmann algebra over an infinite field F of characteristic different from two. In this paper we study the differential identities of G with respect to the action of a finite dimensional Lie algebra L of inner derivations. We explicitly determine a set of generators of the ideal of differential identities of G. Also in case F is of characteristic zero, we study the space of multilinear differential identities in n variables as a module for the symmetric group Sn and we compute the decomposition of the corresponding character into irreducibles. Finally, we prove that unlike the ordinary case the variety of differential algebras with L action generated by G has no almost polynomial growth.

Determinants and characteristic polynomials of Lie algebras

For an s-tuple A=(A1,…,As) of square matrices of the same size, the (joint) determinant of A and the characteristic polynomial of A are defined bydet⁡(A)(z)=det⁡(z1A1+z2A2+⋯+zsAs) andpA(z)=det⁡(z0I+z1A1+z2A2+⋯+zsAs), respectively. This paper calculates determinant of the finite dimensional irreducible representations of sl(2,F), which is either zero or a product of some irreducible quadratic polynomials. Moreover, it shows that a finite dimensional Lie algebra is solvable if and only if the characteristic polynomial is completely reducible.

Analysis of the symmetry group and exact solutions of the dispersionless KP equation in n + 1 dimensions

The Lie algebra of the symmetry group of the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili equation is obtained and identified as a semi-direct sum of a finite dimensional simple Lie algebra and an infinite dimensional nilpotent subalgebra. Group transformation properties of solutions under the subalgebra sl(2,R) are presented. Known explicit analytic solutions in the literature are shown to be actually group-invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group.

Local automorphisms of finite dimensional simple Lie algebras

Let g be a finite dimensional simple Lie algebra over an algebraically closed field K of characteristic 0. A linear map φ:g→g is called a local automorphism if for every x in g there is an automorphism φx of g such that φ(x)=φx(x). We prove that a linear map φ:g→g is local automorphism if and only if it is an automorphism or an anti-automorphism.

$$\mathsf {Lie}$$Lie-Isoclinism of Pairs of Leibniz Algebras

The aim of this paper is to consider the relation between $$\mathsf {Lie}$$-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional Lie algebras, these concepts are not identical, even if the pairs of Leibniz algebras are $$\mathsf {Lie}$$-stem. Moreover, throughout the paper, we provide some conditions under which $$\mathsf {Lie}$$-isoclinism and isomorphism of $$\mathsf {Lie}$$-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well.

The varieties of semi-conformal vectors of affine vertex operator algebras

This is a continuation of our work to understand vertex operator algebras using the geometric properties of the varieties of semi-conformal vectors attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras associated to a finite dimensional simple Lie algebra g, we describe the variety of semi-conformal vectors by some matrix equations. For general cases, these matrix equations are more subtle to solve. However, for an affine vertex operator algebra associated to g, we find the adjoint group G of g acts on the corresponding variety by adjoint representation of G on g. Thus the G-orbit structures of this variety will provide information of semi-conformal vertex operator subalgebras. Based on this approach, as an example, we consider affine vertex operator algebras associated to the Lie algebra sl2(C). We shall give the decomposition of G-orbits of the variety of the semi-conformal vectors according to different levels. Our results imply that such orbit structures depend on the levels of affine vertex operator algebras.

Multiple Killing horizons

Killing horizons which can be such for two or more linearly independent Killing vectors are studied. We provide a rigorous definition and then show that the set of Killing vectors sharing a Killing horizon is a Lie algebra of dimension at most the dimension of the spacetime. We prove that one cannot attach different surface gravities to such multiple Killing horizons, as they have an essentially unique nonzero surface gravity (or none). always contains an Abelian (sub)algebra—whose elements all have vanishing surface gravity—of dimension equal to or one less than dim . There arise only two inequivalent possibilities, depending on whether or not the nonzero surface gravity exists. We show the connection with near-horizon geometries, and also present a linear system of PDEs, the master equation, for the proportionality function on the horizon between two Killing vectors of a multiple Killing horizon, with its integrability conditions. We provide explicit examples of all possible types of multiple Killing horizons, as well as a full classification of them in maximally symmetric spacetimes.

Quantum Fisher information for unitary parametrization processes governed by arbitrary Lie algebras

Quantum Fisher information (QFI) is a key concept of quantum metrology, which is widely used in quantum parameter estimation. We introduce a general approach to analytically get the QFI of pure states for unitary parametrization processes governed by arbitrary finite dimensional Lie algebras. We show the universality of this approach by comparing the result of algebra with the previous work and applying it to and algebras. For case, there are some interesting findings. The phase of the initial coherent state has an optimal value for maximum QFI, and the QFI can vanish and revive during the time evolution.

Higher-dimensional kinematical Lie algebras via deformation theory

We classify kinematical Lie algebras in dimension D + 1 for D > 3 up to isomorphism. This is part of a series of papers applying deformation theory to the classification of kinematical Lie algebras in arbitrary dimension. This is approached via the classification of deformations of the relevant static kinematical Lie algebra. We also classify the deformations of the universal central extension of the static kinematical Lie algebra in dimension D + 1 for D > 3. In addition, we determine which of these Lie algebras admit an invariant inner product.

Kinematical lie algebras in 2 + 1 dimensions

We classify kinematical Lie algebras in dimension 2 + 1. This is approached via the classification of deformations of the static kinematical Lie algebra. In addition, we determine which kinematical Lie algebras admit invariant symmetric inner products.

Relations in Quantized Function Algebras

We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie algebras.

Foundations of the Theory of Dual Lie Algebras

In this paper we introduce and study dual Lie algebras, i.e., Lie algebras over the algebra of dual numbers. We establish some fundamental properties of such Lie algebras and compare them with the corresponding properties of real and complex Lie algebras. We discuss the question of classification of dual Lie algebras of small dimension and consider the connection of dual Lie algebras with approximate Lie algebras.

Rota-Baxter operators of 3-dimensional Heisenberg Lie algebra

In this paper, we consider the question of the Rota-Baxter operators of 3-dimensional Heisenberg Lie algebra on $\mathbb{F}$, where $\mathbb{F}$ is an algebraic closed field. By using the Lie product of the basis elements of Heisenberg Lie algebras, all Rota-Baxter operators of 3-dimensional Heisenberg Lie algebras are calculated and left symmetric algebras of 3-dimensional Heisenberg Lie algebra are determined by using the Yang-Baxter operators.