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Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra and set by $\tilde{\mathfrak{g}}:=\mathfrak{g}\otimes_{\mathbb C} \mathbb{C}[t,t^{-1}]$ its loop algebra.

How to express the universal enveloping algebra $U(\tilde{\mathfrak{g}})$ of $\tilde{\mathfrak{g}}$ as a quotient $\frac{ A_{X}}{I}$, where $A_{X}$ is a free associative algebra over some set $X$ and $I$ is an ideal of $A_{X}$?

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Any algebra whatsoever is a quotient of a free associative algebra (take any set of generators, e.g. all elements of the algebra, and all relations between them). And you mean $U(\tilde{\mathfrak{g}})$, right? –  Qiaochu Yuan Aug 6 '12 at 3:17
You mean that I have to take all elements $x^\pm_\alpha\otimes t^r$, where $\alpha$ is a positive root of $\mathfrak{g}$ and $r\in \mathbb Z$, with ALL relations between these elements, is it? How to take a minimal and explicit set of relations in this case? –  Matt Elly Aug 6 '12 at 3:23
That is a very different question. The universal enveloping algebra comes with a distinguished presentation that is not a bad idea to use. –  Qiaochu Yuan Aug 6 '12 at 4:10
Off topic note: There is a very prominent math.SE user whose name is Matt E (no period) math.stackexchange.com/users/221/matt-e You might want to avoid confusion by choosing a more distinctive name. Welcome to math.SE! –  David Speyer Aug 6 '12 at 11:31
@David Speyer: So sorry! I guess it is already fixed. Thanks. –  Matt Elly Aug 6 '12 at 13:05

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