Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press, Inc. iv, 184 p. (1991).
Consider a differential equation , with homogeneous quadratic, on a finite dimensional vector space . Define a bilinear composition on by , Then with this multiplication, is a commutative, nonassociative algebra, and a linear, solution-preserving map between two differential equations of the type above is a homomorphism of the corresponding algebras.
The first part (1-6) contains the fundamental theory and its application to some classes of examples. The basic definitions and properties are presented; then subalgebras and algebraic invariant subsets are studied with emphasis on semi-invariants.
In the second part (7-10), the objects of investigation are subalgebras of the Lie algebra Pol of all polynomial vector fields on a vector space . If a given homogeneous polynomial is contained in a large finite dimensional subalgebra of Pol or if its centralizer is large, it implies conditions on the algebra determined by that can be very strong. This connection between the inner structure of and its role in Pol is the main subject of interest. The transitive subalgebra and the differential equations related to it are discussed, and the centralizer of a given homogeneous element of Pol is studied. Classes of differential equations can thus be solved explicitly.