This is my context: I have a simple Lie algebra $\mathfrak{g}$ defined over $\mathbb{C}$ , and the corresponding affine Kac-Moody algebra $\mathfrak{g}_{k}$ , defined as the central extension by $\mathbb{C}\textbf{1}$ of $\mathfrak{g}((t)) = \mathfrak{g}\otimes \mathbb{C}((t))$ , where $\mathbb{C}((t))$ denotes formal Laurent power series. What we are looking for are representations of $\mathfrak{g}_{k}$ such that
- $\textbf{1}$ acts as the identity
- for every $v\in V$ there exists $N\geq 0$ such that $(\mathfrak{g}\otimes t^N \mathbb{C}[[t]])\cdot v = 0$
We want to associate an enveloping algebra $\tilde{\mathcal{U}}(\mathfrak{g})$ to $\mathfrak{g}$ such that the aforementioned representations of $\mathfrak{g}$ are exactly the representations of $\tilde{\mathcal{U}}(\mathfrak{g})$ as an associative algebra (i.e. left $\tilde{\mathcal{U}}(\mathfrak{g})$-modules). Starting from the universal enveloping algebra $\mathcal{U}(\mathfrak{g}_k)$ we first quotient by $\textbf{1}-1$, so that $\textbf{1}$ acts as the identity. We call this algebra $\mathcal{U}(\mathfrak{\hat{g}}_k)$. This is fine to me.
Now the problem: for the second condition to hold, we define a linear topology on $\mathcal{U}(\mathfrak{\hat{g}}_k)$ by using ad the basis of neighborhoods for $0$ the following left ideals:
$$I_n = \mathcal{U}(\mathfrak{\hat{g}}_k) \cdot (\mathfrak{g}\otimes t^N \mathbb{C}[[t]])$$ and then we define $\tilde{\mathcal{U}}(\mathfrak{g}_k)$ as the completion of $\mathcal{U}(\hat{\mathfrak{g}}_k)$ with respect to this topology, or equivalently as the inverse limit $$\tilde{\mathcal{U}}(\mathfrak{g}) = \lim_{\longleftarrow}\frac{\mathcal{U}(\hat{\mathfrak{g}}_k)}{I_N}$$ What does completion mean in this context?
For reference: I am quoting section 2.1.2 of this book. I have a couple of questions: first of all, what are those quotients? Only vector spaces? And why are we allowed to quotient by left ideals?
The book states "Even though the $I_N$ are only left ideals, one checks that the associative product structure on $\mathcal{U}(\mathfrak{g})$ extends by continuity to an associative product structure on $\tilde{\mathcal{U}}(\mathfrak{g})$".
I guess that I don't really understand what it means to give a topology in this way, so I would be really glad if someone could shed some light on that, and then explain why the algebra $\tilde{\mathcal{U}}(\mathfrak{g})$ has an associative product structure.
Moreover, later on the author says that an element of $\tilde{\mathcal{U}}(\mathfrak{sl}_2)$ is of the form $$K + \sum_{n\geq 0} Q_n e_n + R_n f_n + S_n h_n$$ where $\{e,h,f\}$ is the standard basis of $\mathfrak{sl}_2$, $e_n$ is defined as $e\otimes t^n$ (same for $f_n$ and $h_n$), where $K, Q_n, R_n, S_n$ are finite linear combinations of monomials in the generators $e_m, f_m, h_m, m \in \mathbb{Z}$. I also tried to see why this must be the case, but I couldn't understand it. I am sure it is related to the definition of inverse limit, i.e. its elements are sequences in the direct product of those quotients which satisfy some compatibility rules, but I can't understand why.