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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?2012-08-06
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    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?2012-08-06
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    That is a very different question. The universal enveloping algebra comes with a distinguished presentation that is not a bad idea to use.2012-08-06
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    Off topic note: There is a very prominent math.SE user whose name is Matt E (no period) http://math.stackexchange.com/users/221/matt-e You might want to avoid confusion by choosing a more distinctive name. Welcome to math.SE!2012-08-06
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    @David Speyer: So sorry! I guess it is already fixed. Thanks.2012-08-06
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    @Qiaochu Yuan: I am still thinking about this situation. Of course your first post answers this not very well formulated question, but even in the case of my last comment I don't know what is the best way to proceed. Do you have some idea about how to do it? I mean: what should be the set of relations? It is very common to see people doing this kind of proceeding for Kac-Moody algebras, since they are given by generators and relations, and as we can construct these algebras starting with loop algebras, I'd like to construct the same thing for them. It was my initial motivation.2012-08-19
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    @Matt: as I said above, the universal enveloping algebra has a standard presentation you can use in terms of a presentation for $\tilde{\mathfrak{g}}$, and $\tilde{\mathfrak{g}}$ in turn has a reasonable presentation in terms of a presentation of $\mathfrak{g}$.2012-08-19

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