This is an upper bound of the dimension of electrical

Lie algebra.

When one calculates the symmetries of a given differential equation, one finds the generators in the form of vector fields and then computes the Lie brackets to get the structure constants of the particular

Lie algebra one has found.

For a Cartan involution [theta] of G, we write g = l + p for the corresponding Cartan decomposition of the

Lie algebra g of G and set K := [G.sup.[theta]] = {g [member of] G : [theta](g) = g}.

A Rota-Baxter operator (of weight zero) on a

Lie algebra (g, [*,*]) is a linear operator P : g [right arrow] g such that

The ten independent elements of the SO(5)

Lie algebra can be identified with (see (6a)-(6c)) the three components of the generator of total spin [??], the total charge operator Q, and the six real and imaginary components of the [??] operator (which "rotates" the AF phase into the SC phases and vice versa).

If (g, [,]) is a

Lie algebra, then (g, [,,]) is a Lts, where [x, y, z] := [[x, y], z] (see [1, 4, 5]).

In this case the

Lie algebra g of G has an Ad(K)-invariant decomposition g = [??] [direct sum] m, where m [subset] g is a linear subspace of g and [??] is the

Lie algebra of K.

Recently,

Lie algebra approach was used to study susceptible-infected-susceptible (SIS) epidemic models [10, 11].

The symmetry generators [V.sub.k] in (6a) associated with each solution [s.sub.k] are often called solution symmetries and generate the abelian

Lie algebra [A.sub.n], while [W.sub.y] is referred to as the homogeneity symmetry.

where [lambda] is the spectral parameter and u is a dependent variable, based on a matrix loop algebra [??] associated with a given matrix

Lie algebra g, often being simple.

Let g be a simple

Lie algebra over C with Dynkin diagram [delta].

In this case (4) admits one-dimensional

Lie algebra spanned by the base vector