Journal of Lie Theory 26 (2016), No. 1, 169--179
Copyright Heldermann Verlag 2016
Representing Lie Algebras Using Approximations with Nilpotent Ideals
Wolfgang Alexander Moens
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
wolfgang.moens@univie.ac.at
[Abstract-pdf]
We prove a refinement of Ado's theorem: a $d$-dimensional nilpotent Lie algebra over an algebraically closed field of characteristic zero with an ideal of class $\varepsilon_1$ and codimension $\varepsilon_2$ admits a faithful representation of degree ${d + \varepsilon_1\choose\varepsilon_1} \cdot {d + \varepsilon_2\choose\varepsilon_2}$. We then apply the theory of almost-algebraic hulls to generalise this result to the representation of arbitrary finite-dimensional Lie algebras and of Lie algebras graded by an abelian, finitely-generated, torsion-free group.
Keywords: Lie algebra, representation, universal enveloping algebra, almost-algebraic Lie algebra, grading.
MSC: 17B35
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