Central extensions of topological current algebras.

*(English)*Zbl 1045.17008
Strasburger, Aleksander (ed.) et al., Geometry and analysis on finite- and infinite-dimensional Lie groups. Proceedings of the workshop on Lie groups and Lie algebras, Bȩdlewo, Poland, September 4–15, 2000. Warszawa: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 55, 61-76 (2002).

When continuous cohomology of Lie algebras came up in the 1970s in the work of Gelfand-Fuchs, people tended to evacuate as much as possible the functional analytic tools and to concentrate on the algebraic side. This may have its origin in the lack of a usefull homological algebra framework of Lie algebra cohomology in some category of topological vector spaces. In recent years, K.-H. Neeb and collaborators reconsider classical folklore subjects as central extensions of (possibly infinite-dimensional) Lie groups and Lie algebras giving functional analysis its natural place.

In this context, the article under review reconsiders the second continuous cohomology space of current algebras, i.e. of Lie algebras of the form \({\mathfrak g}_A=A\otimes{\mathfrak g}\) where \(A\) is a (unital associative commutative) \({\mathbf K}\)-algebra and \({\mathfrak g}\) a finite-dimensional semisimple Lie algebra. Here \({\mathbf K}\) means some category of locally convex topological vector spaces (as there is no ‘best’ one, the author considers them as a variable). One can think of Fréchet spaces for definiteness.

For \(A={\mathcal C}^{\infty}(M)\) for a compact manifold \(M\), the answer is known: the center of the universal central extension of \({\mathfrak g}_A\) is \(\Omega^1(M)/d\Omega^0(M)\), all \(1\)-forms divided by exact one’s, as shown in [A. Pressley and G. Segal, Loop groups (1986; Zbl 0618.22011), prop. (4.2.8)]. The relevance of Maier’s article resides in the focus on the universal differential \({\mathbf K}\)-module \((\Omega^1_A,d_A)\) for the \({\mathbf K}\)-algebra \(A\) (\(d_A:A\to\Omega^1_A\) is the universal continuous derivation). Its introduction in Pressley-Segal’s scheme of proof provides both generalization and clarification.

Let me sketch very roughly the proof of the main theorem, namely that the center of the universal central extension of \({\mathfrak g}_A\) is \(\Omega^1_A/d_AA\) in the particular case of a simple \({\mathfrak g}\). As \({\mathfrak g}_A\) is perfect, it is enough to show that each continuous \(2\)-cocycle \(\omega\) factors over the universal cocycle \(\omega_A\) given by \(\omega_A(a\otimes x, b\otimes y)=[ad_Ab]\kappa(x,y)\) where \([ad_Ab]\) is the class of \(ad_Ab\) in \(\Omega^1_A/d_AA\) and \(\kappa\) is the Killing form of \({\mathfrak g}\). Semi-simplicity of the \({\mathfrak g}\)-action on the continuous cochain spaces implies that \(\omega\) can be supposed to be \({\mathfrak g}\)-invariant. The cocycle property together with \({\mathfrak g}\)-invariance constitute strong constraints on \(\omega\), for example, it follows purely algebraically that \(\omega(1\otimes{\mathfrak g},{\mathfrak g}_A)=0\). On the other hand, fixing the \(A\)-arguments in \(\omega\) gives an invariant bilinear form on \({\mathfrak g}\) which must be a multiple of the Killing form. It follows that \[ \omega(a\otimes x, b\otimes y)=\eta(a,b)\kappa(x,y) \] with \(a,b\in A\) and \(x,y\in{\mathfrak g}\), for some continuous skew-symmetric bilinear form \(\eta\) on \(A\). Exploiting the cocycle identity of \(\omega\) and the above-mentioned property, one shows that \(\eta\) factors over \(\Omega^1_A/d_AA\) by universality of \(\Omega^1_A\).

This article gives proofs of some more well-known statements whose proof is hard to find in the literature like the statement \({\mathcal C}^{\infty}(M\times N)\cong {\mathcal C}^{\infty}(M)\widehat{\otimes}{\mathcal C}^{\infty}(N)\) where \(M\), \(N\) are arbitrary smooth manifolds, \(\widehat{\otimes}\) is the completed projective tensor product and \(\cong\) an isomorphism of Fréchet spaces. Furthermore, the author opens up the search for identification of \(\Omega^1_A\) for locally convex topological algebras \(A\) arising in geometric contexts (depending on the category \({\mathbf K}\)). He does this identification for \(A={\mathcal C}^{\infty}(M)\) with the \({\mathcal C}^{\infty}\) topology and \(M\) arbitrary smooth (result: \(\Omega^1_A\cong \Omega^1(M)\)), \(A={\mathcal C}^{\infty}_c(M)\) (smooth functions with compact support) with the LF-topology (result: \(\Omega^1_{A+1\cdot{\mathbb R}}\cong \Omega^1_c(M)\)) and for \(A={\mathcal C}^0(X)\), \(X\) being a compact topological space (result: \(\Omega^1_{A}=0\)). Further research in this direction includes work of Neeb on compact manifolds with boundary, and K.-H. Neeb and F. Wagemann [Manuscr. Math. 112, 441–458 (2003; Zbl 1071.17021)] on current algebras of holomorphic maps on Riemannian domains over Stein manifolds.

Still another interesting question is the passage to group level, i.e. to find (universal central) extensions of the corresponding current groups. The general criteria for the existence are due to K.-H. Neeb [Ann. Inst. Fourier 52, 1365–1442 (2002; Zbl 1019.22012)] and existence in the case of current groups is discussed by P. Maier and K.-H. Neeb [Math. Ann. 326, 367–415 (2003; Zbl 1029.22025)].

For the entire collection see [Zbl 0989.00033].

In this context, the article under review reconsiders the second continuous cohomology space of current algebras, i.e. of Lie algebras of the form \({\mathfrak g}_A=A\otimes{\mathfrak g}\) where \(A\) is a (unital associative commutative) \({\mathbf K}\)-algebra and \({\mathfrak g}\) a finite-dimensional semisimple Lie algebra. Here \({\mathbf K}\) means some category of locally convex topological vector spaces (as there is no ‘best’ one, the author considers them as a variable). One can think of Fréchet spaces for definiteness.

For \(A={\mathcal C}^{\infty}(M)\) for a compact manifold \(M\), the answer is known: the center of the universal central extension of \({\mathfrak g}_A\) is \(\Omega^1(M)/d\Omega^0(M)\), all \(1\)-forms divided by exact one’s, as shown in [A. Pressley and G. Segal, Loop groups (1986; Zbl 0618.22011), prop. (4.2.8)]. The relevance of Maier’s article resides in the focus on the universal differential \({\mathbf K}\)-module \((\Omega^1_A,d_A)\) for the \({\mathbf K}\)-algebra \(A\) (\(d_A:A\to\Omega^1_A\) is the universal continuous derivation). Its introduction in Pressley-Segal’s scheme of proof provides both generalization and clarification.

Let me sketch very roughly the proof of the main theorem, namely that the center of the universal central extension of \({\mathfrak g}_A\) is \(\Omega^1_A/d_AA\) in the particular case of a simple \({\mathfrak g}\). As \({\mathfrak g}_A\) is perfect, it is enough to show that each continuous \(2\)-cocycle \(\omega\) factors over the universal cocycle \(\omega_A\) given by \(\omega_A(a\otimes x, b\otimes y)=[ad_Ab]\kappa(x,y)\) where \([ad_Ab]\) is the class of \(ad_Ab\) in \(\Omega^1_A/d_AA\) and \(\kappa\) is the Killing form of \({\mathfrak g}\). Semi-simplicity of the \({\mathfrak g}\)-action on the continuous cochain spaces implies that \(\omega\) can be supposed to be \({\mathfrak g}\)-invariant. The cocycle property together with \({\mathfrak g}\)-invariance constitute strong constraints on \(\omega\), for example, it follows purely algebraically that \(\omega(1\otimes{\mathfrak g},{\mathfrak g}_A)=0\). On the other hand, fixing the \(A\)-arguments in \(\omega\) gives an invariant bilinear form on \({\mathfrak g}\) which must be a multiple of the Killing form. It follows that \[ \omega(a\otimes x, b\otimes y)=\eta(a,b)\kappa(x,y) \] with \(a,b\in A\) and \(x,y\in{\mathfrak g}\), for some continuous skew-symmetric bilinear form \(\eta\) on \(A\). Exploiting the cocycle identity of \(\omega\) and the above-mentioned property, one shows that \(\eta\) factors over \(\Omega^1_A/d_AA\) by universality of \(\Omega^1_A\).

This article gives proofs of some more well-known statements whose proof is hard to find in the literature like the statement \({\mathcal C}^{\infty}(M\times N)\cong {\mathcal C}^{\infty}(M)\widehat{\otimes}{\mathcal C}^{\infty}(N)\) where \(M\), \(N\) are arbitrary smooth manifolds, \(\widehat{\otimes}\) is the completed projective tensor product and \(\cong\) an isomorphism of Fréchet spaces. Furthermore, the author opens up the search for identification of \(\Omega^1_A\) for locally convex topological algebras \(A\) arising in geometric contexts (depending on the category \({\mathbf K}\)). He does this identification for \(A={\mathcal C}^{\infty}(M)\) with the \({\mathcal C}^{\infty}\) topology and \(M\) arbitrary smooth (result: \(\Omega^1_A\cong \Omega^1(M)\)), \(A={\mathcal C}^{\infty}_c(M)\) (smooth functions with compact support) with the LF-topology (result: \(\Omega^1_{A+1\cdot{\mathbb R}}\cong \Omega^1_c(M)\)) and for \(A={\mathcal C}^0(X)\), \(X\) being a compact topological space (result: \(\Omega^1_{A}=0\)). Further research in this direction includes work of Neeb on compact manifolds with boundary, and K.-H. Neeb and F. Wagemann [Manuscr. Math. 112, 441–458 (2003; Zbl 1071.17021)] on current algebras of holomorphic maps on Riemannian domains over Stein manifolds.

Still another interesting question is the passage to group level, i.e. to find (universal central) extensions of the corresponding current groups. The general criteria for the existence are due to K.-H. Neeb [Ann. Inst. Fourier 52, 1365–1442 (2002; Zbl 1019.22012)] and existence in the case of current groups is discussed by P. Maier and K.-H. Neeb [Math. Ann. 326, 367–415 (2003; Zbl 1029.22025)].

For the entire collection see [Zbl 0989.00033].

Reviewer: Friedrich Wagemann (Nantes)

##### MSC:

17B56 | Cohomology of Lie (super)algebras |

17B65 | Infinite-dimensional Lie (super)algebras |

46H70 | Nonassociative topological algebras |

18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |