Ideal spanned by monomial of degree $n$ in the Universal Enveloping Algebra

Question

I'm trying to understand the Ado theorem proof which uses the universal enveloping algebra of a Lie algebra. In this proof we use the ideal spanned by all monomial of degree $n$ in the universal envoloping algebra. I'm wondering if it's possible that this ideal contains monomial of degree less than $n$, and if so why is it an ideal?

Answer 1

We can use an order function on the monomials to show that it is an ideal.

Let $\mathfrak{g}$ be a finite-dimensional (w.l.o.g. nilpotent) Lie algebra over a field of characteristic zero. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. By the Poincare-Birkhoff-Witt Theorem the ordered monomials $$X^{\alpha}=X_1^{\alpha_1} \ldots X_n^{\alpha_n}, \; \alpha=(\alpha_1,\ldots \alpha_n)\;\in \; \mathbb{Z}_+^n$$ form a basis for $U(\mathfrak{g})$. Let $T=\sum_{\alpha}c_{\alpha}X^{\alpha}$ be an element of $U(\mathfrak{g})$ (with only finitely many nonzero $c_{\alpha}$). Define an order function as follows: $$\matrix{o (X_j) &:= &\max \{m:\;X_j \;\in \; \mathfrak{g}^{(m)}\} & \;\; & o (X^{\alpha}):=\sum_{j=1}^n \alpha_j o (X_j) \cr o (T) & :=& \min \{o (X^{\alpha}) \;: \; c_{\alpha} \neq 0\}\hfill & & o (1_{U(\mathfrak{g})})=0,\; o (0)= \infty \hfill\cr}$$

One can show that the order function satisfies:

$$\matrix{o (T_1+ \cdots + T_j)&\ge &\min \{o (T_1),\ldots,o(T_j)\} \cr o (T_1 \ldots T_j) \hfill & \ge & o (T_1) + \cdots + o (T_j) \hfill\cr}$$

Now let $$U^m (\mathfrak{g})=\{T \; \in \; U(\mathfrak{g}) \; : \; o (T) \ge m \}$$ From the above it is clear that $U^m (\mathfrak{g})$ is an ideal of $U(\mathfrak{g})$ having finite codimension.