Yet another proof of the Ado theorem

Abstract

We give a simple proof of the Birkhoff theorem about existence of a faithful representation for any finite-dimensional nilpotent Lie algebra of characteristic zero.

Ado theorem for nilpotent Hom–Lie algebras

“…Proof. The proof repeats verbatim those of [Z,Lemma 2.7]. If each ρ x is nilpotent, or multiplicative, or nondegenerate, the resulting representation, being assembled as a direct sum of nilpotent, or multiplicative, or nondegenerate representations, is itself nilpotent, or multiplicative, or nondegenerate.…”

“…Finally, we arrive at our main result, whose proof is assembled from the previous lemmas exactly in the same way as the proof of [Z,Theorem 2.11].…”

“…The following lemma, of combinatorial character, is an almost verbatim repetition of [Z,Lemma 2.10] which treats the Lie case, and shows that one can always distinguish elements of a finite-dimensional nilpotent Hom-Lie algebra by kernels of suitable representations.…”

Special and exceptional mock-Lie algebras

“…Completely similar with [Zu,Lemma 1.3], we have an elementary Lemma 4. Let L be a mock-Lie algebra, V an L-module, and D an antiderivation of L with values in V such that Ker D = 0.…”

“…Thus we have only a very limited analog of [Zu,Lemma 2.5], with N <4 -gradings instead of arbitrary N-gradings: Lemma 6. An N <4 -graded mock-Lie algebra has a faithful representation.…”