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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 $x'=p(x)$, with $p$ homogeneous quadratic, on a finite dimensional vector space $V$. Define a bilinear composition on $V$ by $xy={1\over 2} (p(x+y)-p(x)-p(y))$, 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. For a second review of this book see [Zbl 0754.17002].

17A01General theory of nonassociative algebra
17-01Textbooks (nonassociative rings and algebras)
34-01Textbooks (ordinary differential equations)
34A34Nonlinear ODE and systems, general
34C20Transformation and reduction of ODE and systems, normal forms
17B66Lie algebras of vector fields and related (super)algebras
17A36Automorphisms, derivations, other operators