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Let us consider a Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $L\mathfrak{g} = C^\infty(S^1, \mathfrak{g})$ the Lie algebra of the loop group $LG$. My question is about continuous Lie algebra 2-cocycles on $L\mathfrak{g}$.

It is well-known (see, e.g., Prop. 4.2.4 in Pressley-Segal "Loop groups") that if $G$ is semisimple and compact, the only continuous $G$-invariant 2-cocycles are of the form $$\omega(X, Y) = \int_{S^1} b(X(t), Y^\prime(t))dt, $$ where $b$ is some (necessarily symmetric) $G$-invariant bilinear map on $\mathfrak{g}$.

If I understand correctly, the proof in Pressley-Segal does not use the compactness assumption (which is a general assumption throughout the entire chapter there), but the compactness is used in the proof that all continuous 2-cocycles on $L\mathfrak{g}$ are cohomologous to a $G$-invariant one (this follows from averaging over $G$, which only works in the compact case).

Question: What is known if $G$ is a non-compact semisimple Lie group?

Still, all continuous $G$-invariant cocycles are of the form given above, but now there may be non-trivial cohomology classes that are not represented by a $G$-invariant one.

More precisely: What are examples of non-trivial classes in $H^2_c(L\mathfrak{g}, \mathbb{R})$ that are not represented by a $G$-invariant one, where $G$ is some semisimple Lie group?

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    $\begingroup$ I think that the Neeb-Wagemann paper arxiv.org/abs/math/0511260 addresses this question (see notably Section 6 — which is Section 7 in the published version). It is also discussed here where it is observed that the (correct) results of Neeb-Wagemann contradict previous work on the subject by Zusmanovich. I think the semisimple case was due to Kassel-Loday, it has significant simplification because of the vanishing of $H_1$ and $H_2$ of the original Lie algebra and surjectivity of the Koszul map. $\endgroup$
    – YCor
    Commented Dec 2, 2022 at 10:14

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Thanks to Yves Cornulier, for suggesting to look at the paper of Neeb and Wagemann. After reading Example 6.2 of that paper (arxiv version), I think the answer to my question is that in fact all 2-cycles have a representative of the form in my original post. In other words, the assumption of $G$-invariance in the proposition from the book of Pressley-Segal can be dropped (at least up to replacing the cocycle with a cohomologous one).

Explicitly, set $A = C^\infty(S^1)$. By the results of Neeb and Wagemann, any cocycle $\omega$ on $L \mathfrak{g} = A \otimes \mathfrak{g}$ is defined by two continuous linear maps, $$f_1 :\Lambda^2 (A) \otimes \mathrm{Sym}^2(\mathfrak{g}) \to \mathbb{R}, \qquad f_2 : A \otimes Z_2(\mathfrak{g}) \to \mathbb{R}.$$ (In general, there is also $f_3$, but that does not occur in this special case.) Moreover, we can also ignore $f_2$, because $H^2(\mathfrak{g}) = 0$ in the semisimple case, hence the $f_2$ part corresponds to a coboundary. The map $f_1$ corresponds to a map $$ \tilde{f}_1: A \times A \to \mathrm{Sym}(\mathfrak{g})^{\mathfrak{g}}$$ and the corresponding cocycle $\omega$ is then given by $$ \omega(X, Y) = \sum_{ij=1}^n \tilde{f}_1(X^i, Y^j)(b_i, b_j),$$ where $b_1, \dots, b_n$ is a basis for $\mathfrak{g}$, and we expanded $X = \sum_i X^i b_i$, $Y = \sum_j Y^j b_j$. Now, the condition on $f_1$ is that there exist a continuous linear map $f_1^\flat : \Omega^1(S^1) \to \mathrm{Sym}^2(\mathfrak{g})^{\mathfrak{g}}$ with $$ \tilde{f}_1(a, b) = f_1^\flat(a \,db - b \,da) $$ and $$ \Gamma(f_1^\flat(da)) = 0$$ for all $a \in A$ (in general there is a term depending on $f_2$ on the right hand side and such cocycles are called "coupled", but this term vanishes in our case). Here $\Gamma: \mathrm{Sym}^2(\mathfrak{g})^{\mathfrak{g}} \to Z_3(\mathfrak{g})$ is the Koszul map, which is injective for semisimple Lie algebras, hence $f_1^\flat(da) = 0$. Hence we are looking for 1-currents $T$ on $S^1$ that are co-closed, $\partial T = 0$, and it is well-known that the space of such is one-dimensional; up to a scalar, they are of the form $$ T(\alpha) = \int_{S^1} \alpha. $$ Hence $f_1^\flat = T \otimes b$ for some $b \in \mathrm{Sym}^2(\mathfrak{g})^{\mathfrak{g}}$ and assembling $\omega$ for this $f_1^\flat$ yields the cocycle from the original post.

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