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Lie algebras connected with associative ones. (English) Zbl 0595.17010
The author presents a purely algebraic approach to different types of infinite dimensional Lie algebras, which is based on natural connections between Lie and some associative algebras. He introduces the notion of Lie bimodule and admissible Lie bimodule, gives some examples and proves the following main Theorem. Let \(M_ i\) be a \(C^{\alpha}\) manifold, and let \(L_ i\) be a Lie algebra of \(C^{\alpha}\) vector fields on \(M_ i\) such that \(C^{\alpha}(M_ i)L_ i\subset L_ i\) \((C^{\alpha}(M)\) denotes the associative algebra of all \(C^{\alpha}\) functions on M) and that there are \(X_ 1,X_ 2,...,X_ n\in L_ i\) with no common zeros, \(i=1,2\). Then a mapping \(s: L_ 1\to L_ 2\) is an isomorphism of the Lie algebras if and only if there is a \(C^{\alpha}\) diffeomorphism \(u: M_ 1\to M_ 2\) such that \(s=u_*\), where \(u_*\) is the natural action of the diffeomorphism u on vector fields.
Reviewer: A.Fleischer
MSC:
17B65 Infinite-dimensional Lie (super)algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
58A99 General theory of differentiable manifolds
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