# Non-invariant forms on loop Lie algebra of semisimple Lie group

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?

• 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.
– YCor
Commented Dec 2, 2022 at 10:14

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.