Generalized relativistic kinematics

We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincaré group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity c) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants R and c. The limit transitions lead to the ordinary (R → ∞) or alternative (c → ∞) but nevertheless relativistic kinematics.     

Unsteady Solutions of Euler Equations Generated by Steady Solutions

Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.     

Infinite dimensional irreducible Lie algebras containing transformations of finite rank

Let be a field of characteristic zero and let V be an infinite dimensional vector space over . A linear transformation x of V is called finitary if . The aim of this paper is to describe irreducible Lie subalgebras of containing nonzero finitary transformations. It turns out that any such algebra is a semidirect product of a finite dimensional Lie algebra and a “dense” Lie subalgebra of for some vector space W. Received January 4, 2000 / Published online March 12, 2001     

Simple left-symmetric algebras with solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.     

Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ²≃ℂ. We describe all representations of \mathfrakg\mathfrak{g} via vector fields in J ⁰ℝ²=ℝ³(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.     

Simple left-symmetric algebras with¶solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied. Received: 10 June 1997 / Revised version: 29 September 1997     

Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.     

The n �� n KdV hierarchy

We introduce two new soliton hierarchies that are generalizations of the KdV hierarchy. Our hierarchies are restrictions of the AKNS n × n hierarchy coming from two unusual splittings of the loop algebra. These splittings come from automorphisms of the loop algebra instead of automorphisms of sl (n, \mathbbC){sl (n, \mathbb{C})} . The flows in the hierarchy include systems of coupled nonlinear Schr?dinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and B?cklund transformations for these hierarchies.     

Complex finitary simple Lie algebras

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra \frak t(V,P)\frak t(V,\mit\Pi );¶ (2) a finitary orthogonal algebra \frak fso (V,q)\frak {fso} (V,q); ¶ (3) a finitary symplectic algebra \frak fsp (V,s)\frak {fsp} (V,s).¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and P\Pi is a subspace of the dual V^* whose annihilator in V is trivial: 0={v ? V | Pv=0}0=\{{v}\in V\mid \Pi {v}=0\}.     

Foliations associated with the structure of a manifold over a Grassmann algebra of even degree exterior forms

In this paper we consider a category of manifolds over the algebra of even degree exterior forms on ℝ ^N . We give examples of suchmanifolds. We explicitly find elements of the pseudogroup of differentiable transformations and demonstrate that on any differentiable manifold there exist affine foliations.     

P ∞ algebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operatorsalgebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operators

The symmetry algebraP _∞=W _∞ ⊕P ⊕I _∞ of integrable systems is defined. As an example, the classical Lie point symmetries of all higher Kadomtsev-Petviashvili equations are obtained. It is shown that of the point symmetries, the (positive) ones belong to theW _∞ symmetries, while the other (negative) ones belong toI _∞ symmetries. The corresponding action on the τ-function is obtained for the positive symmetries. The negative symmetries cannot be obtained from the free fermion algebra. A new embedding of the Virasoro algebra intogl(∞) describes the conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 231–260, November, 1997.     

平面上线性变换的层次

线性变换在线性代数教学中占有重要的地位.借助齐次坐标描述平面上线性变换的矩阵结构和几何特性,分析平面线性变换包含的层次关系.加深学生对线性变换直观理解.     

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ ₃ of \mathbb R³{{\mathbb R^3}} . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.     

Generalized Lorentz transformations

A real representation of Dirac algebra, using η=diag(−1,1,1,1) as standard metric is discussed. Among other interesting properties it allows to define a generalization of Lorentz transformations. Ordinary boosts and rotations are subsets The additional transformations are shown to describe transformations to displaced systems, rotating systems, “charged systems”, and others. Poincaré transformations are shown to be approximations of these generalized Lorentz transformations. Appendix D gives an interpretation. Ubi materia, ibi geometria (Johannes Kepler)     

Bäcklund Algebra and the KdV Hierarchy

This article presents the Korteweg-de Vries hierarchy in the framework of the Lie algebra of B?cklund transformations and points to the problems raised by its vanishing residues characterization. This paper is in final form and no version of it will be submitted for publication elsewhere. Leonardo da Vinci Lecture held on April 26, 2004 Received: February 2005     

Quaternion Fourier Transform on Quaternion Fields and Generalizations

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations. I thank my family and FTHD organizer S.L. Eriksson. Soli Deo Gloria     

The Szeg? Metric Associated to Hardy Spaces of Clifford Algebra Valued Functions and Some Geometric Properties

In analogy to complex function theory we introduce a Szeg? metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in \mathbbR^m+1{\mathbb{R}^{m+1}} . These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szeg? metric turns out to have a pseudo-invariance under M?bius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.     

The local index formula in noncommutative geometry

In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded selfadjoint operatorD, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.We dedicate this paper to Misha Gromov     

Crossed Products for Weak Hopf Algebras

Weak crossed products by a weak bialgebra are defined. The resulting structure is not that of a unital algebra but an associative algebra with preunit. A general formula of such a product is given in terms of a weak 2-cocycle and a weak measuring. The relation with weak cleft extensions is studied. Equivalences of weak crossed products are defined and are related to gauge transformations. A relation between cleaving maps is described in terms of gauge transformations.