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Lsym

Lsym is a shorthand used in some mathematical literatures to denote symmetry-related structures associated with linear or Lie symmetries of a system. It is not a universally fixed term, but in common usage it refers to symmetry entities that can be exploited to simplify or solve problems.

In differential equations and mathematical physics, Lsym most often denotes the Lie symmetry algebra of a differential

In linear algebra contexts, Lsym can denote a linear symmetric operator or a symmetric matrix, i.e., a

Computation typically involves solving determining equations for point or contact symmetries, or verifying symmetry of a

Applications include reduction of differential equations, construction of invariant solutions, and, where appropriate, spectral decompositions in

See also Lie group, Lie algebra, symmetry, Noether's theorem, differential equation, invariant.

equation.
This
algebra
consists
of
infinitesimal
generators—vector
fields
on
the
space
of
independent
and
dependent
variables—whose
prolongations
leave
the
equation
invariant.
The
resulting
Lie
algebra
encodes
continuous
symmetries
of
the
equation
and
can
be
used
to
find
invariant
solutions
or
reduce
the
order
of
the
system.
linear
map
L
with
L
=
L^T.
Such
operators
are
diagonalizable
with
real
eigenvalues,
and
they
arise
in
energy
forms,
inner
products,
and
spectral
problems.
The
same
notation
may
appear
in
algorithms
to
emphasize
symmetry
properties
of
matrices.
given
operator.
A
simple
illustration
is
that
the
equation
dy/dx
=
y
has
a
scaling
symmetry
x-translation
combined
with
y-scaling,
generating
a
one-parameter
group
whose
infinitesimal
generator
belongs
to
Lsym
of
the
equation.
linear
problems.