87 research outputs found
Maximal Abelian Subalgebras of e(p,q) algebras
Maximal abelian subalgebras of one of the classical real inhomogeneous Lie
algebras are constructed, namely those of the pseudoeuclidean Lie algebra
e(p,q). Use is made of the semidirect sum structure of e(p,q) with the
translations T(p+q) as an abelian ideal. We first construct splitting MASAs
that are themselves direct sums of abelian subalgebras of o(p,q) and of
subalgebras of T(p+q). The splitting subalgebras are used to construct the
complementary nonsplitting ones. We present general decomposition theorems and
construct indecomposable MASAs for all algebras e(p,q), p \geq q \geq 0. The
case of q=0 and 1 were treated earlier in a physical context. The case q=2 is
analyzed here in detail as an illustration of the general results.Comment: 29 pages, Late
Discrete matrix Riccati equations with superposition formulas
An ordinary differential equation is said to have a superposition formula if
its general solution can be expressed as a function of a finite number of
particular solution. Nonlinear ODE's with superposition formulas include matrix
Riccati equations. Here we shall describe discretizations of Riccati equations
that preserve the superposition formulas. The approach is general enough to
include -derivatives and standard discrete derivatives.Comment: 20 pages; v.2: a misprint correcte
Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
The symmetry algebra of the real elliptic Liouville equation is an
infinite-dimensional loop algebra with the simple Lie algebra as its
maximal finite-dimensional subalgebra. The entire algebra generates the
conformal group of the Euclidean plane . This infinite-dimensional algebra
distinguishes the elliptic Liouville equation from the hyperbolic one with its
symmetry algebra that is the direct sum of two Virasoro algebras. Following a
discretisation procedure developed earlier, we present a difference scheme that
is invariant under the group and has the elliptic Liouville equation
in polar coordinates as its continuous limit. The lattice is a solution of an
equation invariant under and is itself invariant under a subgroup of
, namely the rotations of the Euclidean plane
Lie-point symmetries of the discrete Liouville equation
The Liouville equation is well known to be linearizable by a point
transformation. It has an infinite dimensional Lie point symmetry algebra
isomorphic to a direct sum of two Virasoro algebras. We show that it is not
possible to discretize the equation keeping the entire symmetry algebra as
point symmetries. We do however construct a difference system approximating the
Liouville equation that is invariant under the maximal finite subalgebra SL_x
\lf 2 , \mathbb{R} \rg \otimes SL_y \lf 2 , \mathbb{R} \rg . The invariant
scheme is an explicit one and provides a much better approximation of exact
solutions than comparable standard (non invariant) schemes
Lie point symmetries and ODEs passing the Painlev\'e test
The Lie point symmetries of ordinary differential equations (ODEs) that are
candidates for having the Painlev\'e property are explored for ODEs of order . Among the 6 ODEs identifying the Painlev\'e transcendents only
, and have nontrivial symmetry algebras and that only
for very special values of the parameters. In those cases the transcendents can
be expressed in terms of simpler functions, i.e. elementary functions,
solutions of linear equations, elliptic functions or Painlev\'e transcendents
occurring at lower order. For higher order or higher degree ODEs that pass the
Painlev\'e test only very partial classifications have been published. We
consider many examples that exist in the literature and show how their symmetry
groups help to identify those that may define genuinely new transcendents
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