Abelian extensions of infinite-dimensional Lie groups.

*(English)*Zbl 1079.22018
Travaux mathématiques. Fasc. XV. Luxembourg: Université du Luxembourg, Séminaire de Mathématique (ISBN 2-87971-251-3/pbk). Trav. Math. 15, 69-194 (2004).

This is a comprehensive study of abelian extensions of general infinite-dimensional Lie groups and their relations to abelian extensions of topological Lie algebras. It is a continuation of the author’s earlier work concerning central extensions [Ann. Inst. Fourier 52, 1365–1442 (2002; Zbl 1019.22012)].

To explain the main results of the paper, let \(G\) be a connected Lie group modeled on a locally convex space and \(A\) be an abelian Lie group of the form \(A={\mathfrak a}/\Gamma\), where \({\mathfrak a}\) is a Mackey complete locally convex space and \(\Gamma\leq {\mathfrak a}\) a discrete additive subgroup. The author considers extensions \(1\to A\to \widehat{G}\overset{q}{\to} {G} 1\) of Lie groups such that \(q\) admits smooth local sections. Any such extension determines a smooth \(G\)-module structure on \(A\) (obtained by factoring the conjugation action of \(\widehat{G}\) over \(q\)), and a cohomology class \([f]\in H^2_s(G,A)\), for a suitable \(A\)-valued \(2\)-cocycle \(f: G\times G\to A\) which is smooth on some neighbourhood of \((1,1)\). The extension is determined by \([f]\) up to equivalence, and every cohomology class arises in this way. Differentiating the action, \({\mathfrak a}\) becomes a topological module over the Lie algebra \({\mathfrak g}\) of \(G\). The continuous Lie algebra cohomology classes \([\omega]\in H^2_c({\mathfrak g},{\mathfrak a})\) parameterize the equivalence classes of extensions of topological Lie algebras \({\mathfrak a}\to \widehat{{\mathfrak g}}\to {\mathfrak g}\) such that \(\widehat{{\mathfrak g}}\to {\mathfrak g}\) admits a continuous linear section and \({\mathfrak g}\) induces the given action on \({\mathfrak a}\). Every abelian Lie group extension determines an abelian extension of topological Lie algebras. In terms of cohomology classes, given \([f]\in H^2_s(G,A)\) the corresponding cohomology class \(D_2([f]):=[Df]\in H_c^2({\mathfrak g},{\mathfrak a})\) is obtained via \(Df(x,y):=d^2f(1,1)(x,y)-d^2f(1,1)(y,x)\) for \(x,y\in {\mathfrak g}\). The studies are centered around the question which classes \([\omega]\in H_c^2({\mathfrak g},{\mathfrak a})\) are in the image of \(D_2\), i.e., which Lie algebra extensions \({\mathfrak a}\to {\mathfrak a}\oplus_\omega {\mathfrak g} \to {\mathfrak g}\) “integrate” to corresponding extensions of Lie groups.

The author’s Integrability Criterion (Theorem 6.7) singles out two obstructions for the existence of \([f]\in H^2_s(G,A)\) such that \(D_2([f])=[\omega]\). The first requirement for integrability is that \(q_A\circ \text{per}_\omega\) vanishes, where \(q_A: {\mathfrak a}\to A={\mathfrak a}/\Gamma\) is the quotient map and \(\text{per}_\omega : \pi_2(G)\to {\mathfrak a}\) the period homomorphism. The latter associates with the homotopy class \([\sigma]\) of a piecewise smooth map \(\sigma: {\mathbb S}^2\to G\) the integral \(\int_\sigma\omega^{\text{ eq}}\), where \(\omega^{\text{ eq}}\) is the uniquely determined equivariant \({\mathfrak a}\)-valued smooth \(2\)-form on \(G\) such that \(\omega^{\text{ eq}}_1=\omega\). The second requirement is that the flux homomorphism \(F_\omega: \pi_1(G)\to H^1_c({\mathfrak g}, {\mathfrak a})\) vanishes, which takes the homotopy class \([\gamma]\) of a piecewise smooth closed path \(\gamma\) to the cohomology class of the \(1\)-cocycle \({\mathfrak g}\to {\mathfrak a}\), \(x\mapsto \int_0^1 \gamma(t).\omega(\gamma(t)^{-1}\gamma'(t), \text{Ad}(\gamma(t))^{-1}.x)\, dt\). Various generalizations and related questions are discussed as well. In particular, some information is provided concerning the case where \(A\) (or \(G\)) is not connected.

It is also shown that, for \(G\) smoothly paracompact, a Lie algebra cocycle \(\omega\) integrates to a global smooth group cocycle \(f: G\times G\to A\) if and only if \(\omega^{\text{ eq}}\) is an exact \(2\)-form and \(F_\omega\) vanishes. Many illustrating examples are given. In particular, the author explains how the period map and flux cocycle can be interpreted in geometric terms if \(G\) is the identity component \(\text{Diff}(M)_0^{\text{ op}}\) of the diffeomorphism group of a compact manifold, \(A={\mathfrak a}=C^\infty(M,V)\) with \(V\) a Fréchet space, and the Lie algebra cocycle \(\omega: {\mathcal V}(M)\times {\mathcal V}(M)\to {\mathfrak a}\) is of the form \(\omega(X,Y):=\omega_M(X,Y)\) for a closed \(V\)-valued \(2\)-form \(\omega_M\) on \(M\) (Section 9).

Section 10 generalizes results by V. Ovsienko and C. Roger [Indag. Math., New Ser. 9, No. 2, 277–288 (1998; Zbl 0932.17029)], who determined the cohomology of the \({\mathcal V}({\mathbb S}^1)\)-module \({\mathcal F}_\lambda\) of \(\lambda\)-densities on \({\mathbb S}^1\) for \(\lambda\in {\mathbb R}\) and discussed which Lie algebra cocycles integrate to cocycles on the group \(G=\text{Diff}({\mathbb S}^1)_0\) of orientation preserving diffeomorphisms. The author obtains new proofs for these results on the basis of his general extension theory, and clarifies in addition which Lie algebra cocycles are not integrable to \(G\), but do integrate to group cocycles on the universal covering group \(\widetilde{G}\).

For the entire collection see [Zbl 1065.22001].

To explain the main results of the paper, let \(G\) be a connected Lie group modeled on a locally convex space and \(A\) be an abelian Lie group of the form \(A={\mathfrak a}/\Gamma\), where \({\mathfrak a}\) is a Mackey complete locally convex space and \(\Gamma\leq {\mathfrak a}\) a discrete additive subgroup. The author considers extensions \(1\to A\to \widehat{G}\overset{q}{\to} {G} 1\) of Lie groups such that \(q\) admits smooth local sections. Any such extension determines a smooth \(G\)-module structure on \(A\) (obtained by factoring the conjugation action of \(\widehat{G}\) over \(q\)), and a cohomology class \([f]\in H^2_s(G,A)\), for a suitable \(A\)-valued \(2\)-cocycle \(f: G\times G\to A\) which is smooth on some neighbourhood of \((1,1)\). The extension is determined by \([f]\) up to equivalence, and every cohomology class arises in this way. Differentiating the action, \({\mathfrak a}\) becomes a topological module over the Lie algebra \({\mathfrak g}\) of \(G\). The continuous Lie algebra cohomology classes \([\omega]\in H^2_c({\mathfrak g},{\mathfrak a})\) parameterize the equivalence classes of extensions of topological Lie algebras \({\mathfrak a}\to \widehat{{\mathfrak g}}\to {\mathfrak g}\) such that \(\widehat{{\mathfrak g}}\to {\mathfrak g}\) admits a continuous linear section and \({\mathfrak g}\) induces the given action on \({\mathfrak a}\). Every abelian Lie group extension determines an abelian extension of topological Lie algebras. In terms of cohomology classes, given \([f]\in H^2_s(G,A)\) the corresponding cohomology class \(D_2([f]):=[Df]\in H_c^2({\mathfrak g},{\mathfrak a})\) is obtained via \(Df(x,y):=d^2f(1,1)(x,y)-d^2f(1,1)(y,x)\) for \(x,y\in {\mathfrak g}\). The studies are centered around the question which classes \([\omega]\in H_c^2({\mathfrak g},{\mathfrak a})\) are in the image of \(D_2\), i.e., which Lie algebra extensions \({\mathfrak a}\to {\mathfrak a}\oplus_\omega {\mathfrak g} \to {\mathfrak g}\) “integrate” to corresponding extensions of Lie groups.

The author’s Integrability Criterion (Theorem 6.7) singles out two obstructions for the existence of \([f]\in H^2_s(G,A)\) such that \(D_2([f])=[\omega]\). The first requirement for integrability is that \(q_A\circ \text{per}_\omega\) vanishes, where \(q_A: {\mathfrak a}\to A={\mathfrak a}/\Gamma\) is the quotient map and \(\text{per}_\omega : \pi_2(G)\to {\mathfrak a}\) the period homomorphism. The latter associates with the homotopy class \([\sigma]\) of a piecewise smooth map \(\sigma: {\mathbb S}^2\to G\) the integral \(\int_\sigma\omega^{\text{ eq}}\), where \(\omega^{\text{ eq}}\) is the uniquely determined equivariant \({\mathfrak a}\)-valued smooth \(2\)-form on \(G\) such that \(\omega^{\text{ eq}}_1=\omega\). The second requirement is that the flux homomorphism \(F_\omega: \pi_1(G)\to H^1_c({\mathfrak g}, {\mathfrak a})\) vanishes, which takes the homotopy class \([\gamma]\) of a piecewise smooth closed path \(\gamma\) to the cohomology class of the \(1\)-cocycle \({\mathfrak g}\to {\mathfrak a}\), \(x\mapsto \int_0^1 \gamma(t).\omega(\gamma(t)^{-1}\gamma'(t), \text{Ad}(\gamma(t))^{-1}.x)\, dt\). Various generalizations and related questions are discussed as well. In particular, some information is provided concerning the case where \(A\) (or \(G\)) is not connected.

It is also shown that, for \(G\) smoothly paracompact, a Lie algebra cocycle \(\omega\) integrates to a global smooth group cocycle \(f: G\times G\to A\) if and only if \(\omega^{\text{ eq}}\) is an exact \(2\)-form and \(F_\omega\) vanishes. Many illustrating examples are given. In particular, the author explains how the period map and flux cocycle can be interpreted in geometric terms if \(G\) is the identity component \(\text{Diff}(M)_0^{\text{ op}}\) of the diffeomorphism group of a compact manifold, \(A={\mathfrak a}=C^\infty(M,V)\) with \(V\) a Fréchet space, and the Lie algebra cocycle \(\omega: {\mathcal V}(M)\times {\mathcal V}(M)\to {\mathfrak a}\) is of the form \(\omega(X,Y):=\omega_M(X,Y)\) for a closed \(V\)-valued \(2\)-form \(\omega_M\) on \(M\) (Section 9).

Section 10 generalizes results by V. Ovsienko and C. Roger [Indag. Math., New Ser. 9, No. 2, 277–288 (1998; Zbl 0932.17029)], who determined the cohomology of the \({\mathcal V}({\mathbb S}^1)\)-module \({\mathcal F}_\lambda\) of \(\lambda\)-densities on \({\mathbb S}^1\) for \(\lambda\in {\mathbb R}\) and discussed which Lie algebra cocycles integrate to cocycles on the group \(G=\text{Diff}({\mathbb S}^1)_0\) of orientation preserving diffeomorphisms. The author obtains new proofs for these results on the basis of his general extension theory, and clarifies in addition which Lie algebra cocycles are not integrable to \(G\), but do integrate to group cocycles on the universal covering group \(\widetilde{G}\).

For the entire collection see [Zbl 1065.22001].

Reviewer: Helge Glöckner (Darmstadt)

##### MSC:

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

17B56 | Cohomology of Lie (super)algebras |

17B66 | Lie algebras of vector fields and related (super) algebras |

22E41 | Continuous cohomology of Lie groups |

57T20 | Homotopy groups of topological groups and homogeneous spaces |