We recall that A(n) denotes the abelian Lie algebra of dimension n and the main results of [2, 5, 6, 7, 11, 18, 19] illustrate that many inequalities on dim M(L) become equalities if and only if L splits in the sums of A(n) and of a Heisenberg algebra H(m) (here m [greater than or equal to] 1 is a given integer).
Since N/N [intersection] [PHI](L) [congruent] A(t) is a direct factor of the abelian Lie algebra L/[PHI](L) [congruent] A(s + t) [congruent] A(s) [direct sum] A(t), Lemma 2.2 implies
The dimension of the Schur multiplier of abelian Lie algebras is a classic.
An Abelian Lie algebra
L spanned by the vector fields [Z.sub.1],..., [Z.sub.p] is called a conditional symmetry algebra of the kth order PDE (2.1) if the vector fields [Z.sub.1],..., [Z.sub.p] are tangent to the subvariety S, i.e