Journal of Lie Theory

no. 3
A Remark on Ado's Theorem for Principal Ideal Domains
Journal of Lie Theory, Volume 34 (2024) no. 3, pp. 531-540
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.
DOI: 10.5802/jolt.1348
Classification: 17B10, 17B30, 17B35
Keywords: Ado's Theorem, Lie algebras, degree of representations 
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     author = {A. Zozaya},
     title = {A {Remark} on {Ado's} {Theorem} for {Principal} {Ideal} {Domains}},
     journal = {Journal of Lie Theory},
     pages = {531--540},
     year = {2024},
     volume = {34},
     number = {3},
     doi = {10.5802/jolt.1348},
     zbl = {1571.17015},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1348/}
}
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A. Zozaya. A Remark on Ado's Theorem for Principal Ideal Domains. Journal of Lie Theory, Volume 34 (2024) no. 3, pp. 531-540. doi: 10.5802/jolt.1348

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