Polynomial Realizations of Finite-Dimensional Lie Algebras
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lynomial Realizations of Finite-Dimensional Lie Algebras V. V. Gorbatsevich Received May 26, 2019; in final form, January 29, 2020; accepted January 31, 2020
Abstract. It is proved that any finite-dimensional Lie algebra over an algebraically closed field K of characteristic 0 can be embedded (realized) as a transitive Lie subalgebra of the Lie algebra of polynomial vector fields on the space K n . The same is also proved for arbitrary real Lie algebras and in some other cases. Key words: Lie algebra, realization of Lie algebra, polynomial Lie algebra, Mal tsev splitting. DOI: 10.1134/S0016266320020021
In this paper we consider polynomial Lie algebras, by which we mean Lie algebras of vector fields on the space K n (where K is a field of characteristic 0) with polynomial coefficients (we refer to such vector fields as polynomial). Note that the term polynomial Lie algebra is sometimes used with a different meaning; see, e.g., [1]. To be more precise, we study not the Lie algebra P(K n ) of polynomial vector fields on K n itself but its finite-dimensional Lie subalgebras and the possibility of realizing any finite-dimensional Lie algebras as such subalgebras. The Lie algebra P(K n ) can be treated as the Lie algebra of derivations of the algebra of polynomials in n variables. By P0 (K n ) we denote the Lie subalgebra of P(K n ) consisting of vector fields taking the value 0 at the origin (this is a stationary subalgebra of the Lie algebra P(K n )). We will prove that if the field K is algebraically closed, then any finite-dimensional Lie algebra can be realized as the Lie algebra of transitive polynomial vector fields (for sufficiently large n). The same is proved in the case where K = R. For arbitrary algebraically nonclosed fields K, some particular results are obtained. Polynomial vector fields proper are fairly popular in various domains of mathematics, such as the theory of differential equations, the theory of singularities, and so on. Not every polynomial vector field is complete, i.e., such a field does not always generate a global one-parameter transformation group. However, in local considerations (assuming that the field K is topological, e.g., K = R or C), we can also consider the local transformations corresponding to these vector fields. There is yet another approach to the problem of realizing Lie algebras, which uses formal power series. But here the problem of the existence of realizations of Lie algebras and even pairs of Lie algebras has long been solved (see [2]). We might also consider rational vector fields (forming the Lie algebra of derivations of the field of rational functions) and the corresponding Lie algebras, but we will not discuss these topics here. Definition 1. If vectors of some Lie algebra of vector fields form a basis at the point 0 ∈ K n , then this Lie algebra is said to be transitive. In fact, it is sometimes necessary to consider not only the point 0 but also other points of K n , because even in Sophus Lie’s “standard” (i.e., conventional) classification of transitive Lie algebras in the plane some
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