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edited Mar 6, 2018 at 13:05

j.c.

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The Kac-Moody central extension can be described in terms of algebraic K_2$K_2$. This was first discovered I think by Spencer Bloch in the early '80s. There is a scattered literature that spells this out in different contexts - the main published references I can think of are by Deligne-Brylinski (Central extensions of groups by K_2)Deligne-Brylinski (Central extensions of groups by $K_2$) and the papers it cites by Deligne (in particular Le Symbole ModereLe Symbole Modéré), the papers by Brylinski-MaclaughlinMclaughlin on the Segal-Witten reciprocity law and symbols etc. (I learned of this from the famous unpublished manuscript of Beilinson-Kazhdanthe famous unpublished manuscript of Beilinson-Kazhdan, I think it appears also in later published works of these two individually). Anyway this gives a formula for the Kac-Moody central extension in terms of the tame symbol. Actually one place where the whole story is spelled out beautifully is Kapranov's paper on Eisenstein series and S-dualityKapranov's paper on Eisenstein series and S-duality.

To summarize briefly: $H^4(BG,Z)$ actually consists of algebraic cycle classes, i.e. it's equal to Chow^2 (BG)$Chow^2 (BG)$.Bloch Bloch showed (in the 70s) that this is the same as $H^2(BG,K_2)$ and used this to give a beautiful picture for second Chern classes. Anyway this can be interpreted as central extensions of G$G$ by $K_2$. Now if you're over a local field (Laurent series say) the tame symbol is a kind of residue map, taking K_2$K_2$ of the local field to K_1$K_1$ (iei.e. units) in the residue field. So you can push out the K_2$K_2$ extension to get a $C^*$ extension of the loop group, as desired. While K_2$K_2$ is an intimidating beast, this gives aan explicit formula I think since the tame symbol is explicit... but I'm the wrong person to give you that formula.

BTW for the multiplicative group this ends up giving a POV on Weil reciprocity, that was spelled out by Witten in his gorgeous paper on Grassmannians QFT and Algebraic Curvesgorgeous paper on Grassmannians, QFT and Algebraic Curves, and is explicated in a paper by Brylinski with a related title (Central Extensions and Reciprocity Laws)a paper by Brylinski with a related title (Central Extensions and Reciprocity Laws) and most recently in a very pretty paper of Takhtajana very pretty paper of Takhtajan.

The Kac-Moody central extension can be described in terms of algebraic K_2. This was first discovered I think by Spencer Bloch in the early '80s. There is a scattered literature that spells this out in different contexts - the main published references I can think of are by Deligne-Brylinski (Central extensions of groups by K_2) and the papers it cites by Deligne (in particular Le Symbole Modere), the papers by Brylinski-Maclaughlin on the Segal-Witten reciprocity law and symbols etc (I learned of this from the famous unpublished manuscript of Beilinson-Kazhdan, I think it appears also in later published works of these two individually). Anyway this gives a formula for the Kac-Moody central extension in terms of the tame symbol. Actually one place where the whole story is spelled out beautifully is Kapranov's paper on Eisenstein series and S-duality.

To summarize briefly: $H^4(BG,Z)$ actually consists of algebraic cycle classes, i.e. it's equal to Chow^2 (BG).Bloch showed (in the 70s) that this is the same as $H^2(BG,K_2)$ and used this to give a beautiful picture for second Chern classes. Anyway this can be interpreted as central extensions of G by $K_2$. Now if you're over a local field (Laurent series say) the tame symbol is a kind of residue map, taking K_2 of the local field to K_1 (ie units) in the residue field. So you can push out the K_2 extension to get a $C^*$ extension of the loop group, as desired. While K_2 is an intimidating beast, this gives a explicit formula I think since the tame symbol is explicit.. but I'm the wrong person to give you that formula.

BTW for the multiplicative group this ends up giving a POV on Weil reciprocity, that was spelled out by Witten in his gorgeous paper on Grassmannians QFT and Algebraic Curves, and is explicated in a paper by Brylinski with a related title (Central Extensions and Reciprocity Laws) and most recently in a very pretty paper of Takhtajan.

The Kac-Moody central extension can be described in terms of algebraic $K_2$. This was first discovered I think by Spencer Bloch in the early '80s. There is a scattered literature that spells this out in different contexts - the main published references I can think of are by Deligne-Brylinski (Central extensions of groups by $K_2$) and the papers it cites by Deligne (in particular Le Symbole Modéré), the papers by Brylinski-Mclaughlin on the Segal-Witten reciprocity law and symbols etc. (I learned of this from the famous unpublished manuscript of Beilinson-Kazhdan, I think it appears also in later published works of these two individually). Anyway this gives a formula for the Kac-Moody central extension in terms of the tame symbol. Actually one place where the whole story is spelled out beautifully is Kapranov's paper on Eisenstein series and S-duality.

To summarize briefly: $H^4(BG,Z)$ actually consists of algebraic cycle classes, i.e. it's equal to $Chow^2 (BG)$. Bloch showed (in the 70s) that this is the same as $H^2(BG,K_2)$ and used this to give a beautiful picture for second Chern classes. Anyway this can be interpreted as central extensions of $G$ by $K_2$. Now if you're over a local field (Laurent series say) the tame symbol is a kind of residue map, taking $K_2$ of the local field to $K_1$ (i.e. units) in the residue field. So you can push out the $K_2$ extension to get a $C^*$ extension of the loop group, as desired. While $K_2$ is an intimidating beast, this gives an explicit formula I think since the tame symbol is explicit... but I'm the wrong person to give you that formula.