Journal of Lie Theory 34 (2024), No. 3, 531--540
Copyright Heldermann Verlag 2024
A Remark on Ado's Theorem for Principal Ideal Domains
Andoni Zozaya
Department of Mathematics, University of the Basque Country UPV/EHU, Leioa (Bizkaia), Spain
andoni.zozaya@ehu.eus
Ado's Theorem had been extended to principal ideal domains independently by Churkin and Weigel. They proved that if R is a principal ideal domain of characteristic zero and L is a Lie algebra over R which is also a free R-module of finite rank, then L admits a finite faithful Lie algebra representation over R. We present a quantitative proof of this result, providing explicit bounds on the degree of the Lie algebra representations in terms of the rank as a free module. To achieve it, we generalise an established embedding theorem for complex Lie algebras: any Lie algebra as above embeds in a larger Lie algebra that decomposes as the direct sum of its nilpotent radical and another subalgebra.
Keywords: Ado's Theorem, Lie algebras, degree of representations.
MSC: 17B10, 17B30, 17B35.
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