An envelope for Malcev algebras. (English) Zbl 1077.17027
Summary: We prove that for every Malcev algebra \(M\) there exist an algebra \(U(M)\) and a monomorphism \(\iota :M \to U(M)^-\) of \(M\) into the commutator algebra \(U(M)^-\) such that the image of \(M\) lies in the alternative center of \(U(M)\), and \(U(M)\) is a universal object with respect to such homomorphisms. The algebra \(U(M)\), in general, is not alternative, but it has a basis of Poincaré-Birkhoff-Witt type over \(M\) and inherits some good properties of universal enveloping algebras of Lie algebras. In particular, the elements of \(M\) can be characterized as the primitive elements of the algebra \(U(M)\) with respect to the diagonal homomorphism \(\Delta : U(M) \to U(M) \otimes U(M)\). An extension of Ado-Iwasawa theorem to Malcev algebras is also proved.
MSC:
| 17D10 | Mal’tsev rings and algebras |
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