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Algebras and differential equations. (English) Zbl 0791.17002
Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press, Inc. iv, 184 p. (1991).

Consider a differential equation ${x}^{\text{'}}=p\left(x\right)$, with $p$ homogeneous quadratic, on a finite dimensional vector space $V$. Define a bilinear composition on $V$ by $xy=\frac{1}{2}\left(p\left(x+y\right)-p\left(x\right)-p\left(y\right)\right)$, Then $V$ 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 $V$ of all polynomial vector fields on a vector space $V$. If a given homogeneous polynomial $p$ is contained in a large finite dimensional subalgebra of Pol $V$ or if its centralizer is large, it implies conditions on the algebra determined by $p$ that can be very strong. This connection between the inner structure of $p$ and its role in Pol $V$ 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 $V$ is studied. Classes of differential equations can thus be solved explicitly.

##### MSC:
 17A01 General theory of nonassociative algebra 17-01 Textbooks (nonassociative rings and algebras) 34-01 Textbooks (ordinary differential equations) 34A34 Nonlinear ODE and systems, general 34C20 Transformation and reduction of ODE and systems, normal forms 17B66 Lie algebras of vector fields and related (super)algebras 17A36 Automorphisms, derivations, other operators