Diagonal locally finite Lie algebras and a version of Ado's theorem
For a locally finite Lie algebra \(L\) the question is studied whether or not \(L\) can be embedded into a locally finite associative algebra. When \(L\) is assumed to be finite dimensional, the answer is positive by Ado's theorem. In general, however, the answer is negative, even if \(L\) is assumed to be simple. The author determines simple locally finite Lie algebras for which the locally finite analog of Ado's theorem holds. In that case there exists a nontrivial inductive system for the Lie algebra. The result is that a simple locally finite Lie algebra can be embedded into a locally finite associative algebra iff it is diagonal. The definition of diagonal Lie algebras requires some preparations. First diagonality is defined for abstract Levi subalgebras, i.e., for the set of Levi algebras associated to the local system, then on locally perfect Lie algebras. The author shows that a semisimple locally perfect Lie algebra \(L\) is diagonal iff for each \(x\in L\) there exists a polynomial \(f_x\) such that \(f_x (\text{ad} x)=0\). Furthermore, every simple locally finite Lie algebra \(L\) is locally perfect; and \(L\) is called diagonal if the locally perfect radical is diagonal. Finally, the notion of a Bratelli diagram for locally perfect Lie algebras is introduced. It is proved that \(L\) is diagonal iff it has a diagonal Bratelli diagram. In the last part the author investigates Ado's theorem for locally perfect Lie algebras.
scientific article; zbMATH DE number 1115980