What question did this study set out to answer?

This work aims to extend Ado's Theorem to principal ideal domains by providing explicit representation bounds for Lie algebras.

March 29, 2026Open Access

A Remark on Ado's Theorem for Principal Ideal Domains

Puntos clave

This work aims to extend Ado's Theorem to principal ideal domains by providing explicit representation bounds for Lie algebras.
Generalized an embedding theorem for complex Lie algebras.
Proved that Lie algebras can embed in larger Lie algebras.
Determined bounds on representation degree related to free module rank.
Confirmed that if R is a principal ideal domain of characteristic zero, then L has a finite faithful representation.
Established explicit bounds for the degree of Lie algebra representations based on module rank.
Demonstrated that any suitable Lie algebra embeds in a direct sum of a nilpotent radical and another subalgebra.

Resumen

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.

Preguntar a la IA

Me gusta

Guardar

Ver artículo completo