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Prove that every nilpotent Lie algebra in characteristic zero can be represented by strictly upper triangular matrices. You may use Ado's theorem and consider the case where the field is algebraically closed and the case where the field is arbitrary.
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To prove that every nilpotent Lie algebra over a field of characteristic zero can be represented by strictly upper triangular matrices, we proceed as follows:
Step 1: Apply Ado's Theorem
Ado's theorem guarantees that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful finite-dimensional representation. For a nilpotent Lie algebra $\mathfrak{g}$, the construction of such a representation can be refined to ensure that the image of $\mathfrak{g}$ consists of nilpotent endomorphisms. This refinement involves quotienting the universal enveloping algebra $U(\mathfrak{g})$ by a power of the augmentation ideal $I$, specifically $I^{c+1}$ where $c$ is the nilpotency class of $\mathfrak{g}$. The resulting finite-dimensional module $U(\mathfrak{g})/I^{c+1}$ yields a faithful representation $\rho: \mathfrak{g} \to \mathfrak{gl}(V)$ where each $\rho(x)$ is nilpotent.
Step 2: Apply Engel's Theorem
Engel's theorem states that if all elements of a Lie algebra $\mathfrak{h} \subset \mathfrak{gl}(V)$ are nilpotent endomorphisms, then there exists a basis of $V$ in which every element of $\mathfrak{h}$ is represented by a strictly upper triangular matrix. Since $\rho(\mathfrak{g})$ is a Lie algebra of nilpotent operators (by Step 1), Engel's theorem applies. Thus, there exists a basis for $V$ such that all $\rho(x)$ are strictly upper triangular.
Conclusion
The composition of the faithful representation $\rho$ with a suitable change of basis (from Engel's theorem) embeds $\mathfrak{g}$ into the Lie algebra of strictly upper triangular matrices. This holds over any field of characteristic zero, as Engel's theorem does not require algebraic closure---nilpotent operators can always be strictly triangularized without eigenvalues.
Final Statement
Therefore, every nilpotent Lie algebra in characteristic zero admits a faithful representation by strictly upper triangular matrices, achieved by combining Ado's theorem (to obtain a nilpotent faithful representation) and Engel's theorem (to triangularize it). This result holds regardless of whether the field is algebraically closed. |
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