Finiteness conditions on special Lie algebras (Q1269966)

The author investigates finitely generated special Lie algebras over an arbitrary field \(F\). If a Lie algebra \(L\) over \(F\) is embeddable as an \(F\)-algebra into a Lie algebra of finite dimension over some extension of \(F\) then \(L\) is called representable. It is known that residual finiteness and speciality are necessary conditions for the representability of a finitely generated Lie algebra. In Theorem 1, an example of finitely generated, residually finite and special but not representable Lie algebra is constructed. A variety of Lie algebras \(\mathcal V\) is called locally representable if every finitely generated algebra of \(\mathcal V\) is representable. Theorem 3 of the article states that if any finitely generated residually finite Lie algebra from a variety \(\mathcal V\) over an infinite field \(F\) is representable then \(\mathcal V\) is locally representable. According to Theorem 4, a finitely generated special Lie algebra over an infinite field is Artinian if and only if it is of finite dimension. As a corollary of this statement, the author formulates Theorem 5: if a finitely generated special Lie algebra \(L\) over a field has a maximal subalgebra of finite dimension then \(L\) has finite dimension too.