View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Functional Analysis 194, 347–409 (2002) doi:10.1006/jfan.2002.3942 Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups Helge Glo¨ ckner1 Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918 E-mail: [email protected] Communicated by L. Gross Received September 4, 2001; revised November 15, 2001; accepted December 16, 2001 We characterize the existence of Lie group structures on quotient groups and the existence of universal complexifications for the class of Baker–Campbell–Hausdorff (BCH–) Lie groups, which subsumes all Banach–Lie groups and ‘‘linear’’ direct limit r r Lie groups, as well as the mapping groups CK ðM; GÞ :¼fg 2 C ðM; GÞ : gjM=K ¼ 1g; for every BCH–Lie group G; second countable finite-dimensional smooth manifold M; compact subset SK of M; and 04r41: Also the corresponding test function r r # groups D ðM; GÞ¼ K CK ðM; GÞ are BCH–Lie groups. 2002 Elsevier Science (USA) 0. INTRODUCTION It is a well-known fact in the theory of Banach–Lie groups that the topological quotient group G=N of a real Banach–Lie group G by a closed normal Lie subgroup N is a Banach–Lie group provided LðNÞ is complemented in LðGÞ as a topological vector space [7, 30]. Only recently, it was observed that the hypothesis that LðNÞ be complemented can be omitted [17, Theorem II.2]. This strengthened result is then used in [17] to characterize those real Banach–Lie groups which possess a universal complexification in the category of complex Banach–Lie groups. It is the goal of the present paper to extend these results to more general classes of infinite-dimensional Lie groups, and to discuss interesting examples. We mainly focus on a class of Lie groups which we call ‘‘Baker– Campbell–Hausdorff (BCH)–Lie groups.’’ These are analytic Lie groups in the sense of [13], modelled on arbitrary, not necessarily sequentially complete, locally convex spaces, whose exponential function induces a local 1 Present address: Department of Mathematics, Darmstadt University of Technology, Schlossgartenstr. 7, 64289 Darmstadt, Germany. 347 0022-1236/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved. 348 HELGE GLO¨ CKNER diffeomorphism at 0 and whose group multiplication is given locally by the Campbell–Hausdorff series (see Section 2 for details). Thus, by definition, BCH–Lie groups share some of the most important properties of Banach– Lie groups. Although the class of BCH–Lie groups excludes diffeomorphism groups and other infinite-dimensional Lie groups which behave quite unlike Banach–Lie groups, it subsumes many important examples. In particular, beside Banach–Lie groups, every linear direct limit Lie group is a BCH–Lie group, and so are all groups of mappings or test functions with values in BCH–Lie groups, as will be shown below. In this article, we develop the general Lie theory of BCH–Lie groups, and in particular a general theory of universal complexifications for these groups. The general results are then applied to the specific examples of BCH–Lie groups described before. It turns out that universal complexifications exist for these groups under natural hypotheses, and we find explicit descriptions of the universal complexifications in important cases. We remark that, as opposed to universal complexifications, many authors call a complex Lie group H a ‘‘complexification’’ of a given real Lie group G if G is a subgroup of H and LðHÞ¼LðGÞC (e.g., [25] in the finite- dimensional case, [29, 42] in the infinite-dimensional setting). This point of view is sufficient for many practical purposes. However, already in the finite- dimensional case, many Lie groups do not possess complexifications in the preceding sense; and if such a complexification exists, it need not be uniquely determined. In contrast, a universal complexification of a real Lie group G in a given category A of complex Lie groups is a complex Lie group GC 2 ob A which satisfies a suitable universal property together with a (not necessarily injective) smooth homomorphism gG : G ! GC (cf. [7, 17, 21]). Whenever it exists, GC is unique up to isomorphism. Surpris- ingly, although it is well-known that every finite-dimensional Lie group has a universal complexification in the category of finite-dimensional complex Lie groups [7], and although universal complexifications are a standard topic in textbooks on finite-dimensional Lie groups, universal complex- ifications of infinite-dimensional Lie groups have hardly been studied in the literature. Only recently, the existence question of universal complex- ifications of Banach–Lie groups was raised in [12] and then fully solved in [17]. In particular, it is known now that there are real Banach–Lie groups which do not have universal complexifications in the category of complex Banach–Lie groups. A theory of universal complex- ifications for more general classes of infinite-dimensional Lie groups is still missing in the literature. The present paper intends to close this gap, at least for the class of BCH–Lie groups. In particular, for a real BCH–Lie group G; we shall find a necessary and sufficient condition for the existence of a universal complexification in the category of complex BCH–Lie groups, as well as a sufficient condition for the existence of a universal complexification LIE GROUP STRUCTURES 349 in the category of all complex Lie groups with complex analytic exponential functions. Let us briefly outline the contents of the individual sections now, summarize the main results, and relate them to the literature. We begin with general terminology and basic facts concerning smooth and analytic Lie groups, as well as the required differential calculus (Section 1). In Section 2, we define BCH–Lie groups and develop the basic Lie theory for such groups, for later use.2 Generalizing the recent results concerning quotients of Banach–Lie groups mentioned above, we prove (Corollary 2.21; see Theorem 2.20 for a complex analogue): 0.1. Quotient Theorem. Let G be a real BCH–Lie group and N be a closed, normal subgroup of G. Then the topological quotient group G=N can be given a real BCH–Lie group structure if and only if N is a Lie subgroup of G. If G is a Banach–Lie group here, we obtain a second characterization in the spirit of the solution to Hilbert’s fifth problem: G=N is a Banach–Lie group if and only if it has no small subgroups. It is well known that the group CrðK; GÞ of Cr-mappings on a compact smooth manifold K with values in a finite-dimensional Lie group G is a 1 Banach–Lie group for r 2 N0; and that C ðK; GÞ is a Fre´ chet–Lie group satisfying the axioms of a BCH–Lie group (see, for example, [31]). For infinite-dimensional G modelled on sequentially complete spaces, CðK; GÞ had been made a Lie group in [37]. The groups D1ðM; GÞ of smooth test functions with values in a finite-dimensional Lie group G had already been given a topological group structure making them ‘‘nuclear Lie groups’’ in 1 [1]. For finite-dimensional G; the Lie groups CK ðM; GÞ have been considered in [33, 34] and a direct limit argument has been used there to specify an analytic structure on D1ðM; GÞ:3 Generalizing these construc- tions, we establish results slightly more general than the following (Sections 3 and 4): 0.2. Mapping Groups and Test Function Groups. Suppose that M is a s-compact finite-dimensional smooth manifold, K M a compact subset, N K r r 2 0 [ f1g; and G a smooth (resp., -analytic) Lie group. Then CK ðM; GÞ and DrðM; GÞ can be made smooth (resp., K-analytic) Lie groups in a natural r r K way, with Lie algebras CK ðM; LðGÞÞ and D ðM; LðGÞÞ: IfGisa -analytic r r BCH–Lie group, then so are CK ðM; GÞ and D ðM; GÞ: 2 Several interesting results concerning the closely related class of ‘‘CBH–Lie groups’’ can be found in [43]. For the most part, the material we present is complementary. 3 We should also mention the ‘‘weighted gauge groups’’ (with values in finite-dimensional Lie groups) studied in [5]. 350 HELGE GLO¨ CKNER A smooth Lie group structure on C1ðM; GÞ in the sense of ‘‘convenient differential calculus’’ had already been established in [26], for any smooth Lie group G in that sense. In Sections 5 and 7, we develop a general theory of universal complex- ifications for BCH–Lie groups. As our third main result, we obtain a characterization of those real BCH–Lie groups which possess a universal complexification in the category of complex BCH–Lie groups (cf. Theorem 7.2, Corollary 7.6): 0.3. Complexification Theorem. Given a real BCH–Lie group G, let N be the intersection of all kernels of smooth homomorphisms from G into complex BCH–Lie groups. Then G has a universal complexification GC in the category of complex BCH–Lie groups if and only if N is a Lie subgroup of G and ðLðGÞ=LðNÞÞC is enlargible. In this case, LðGCÞffiðLðGÞ=LðNÞÞC: If GC exists in the category of complex BCH–Lie groups and N is a discrete subgroup of G, then in fact GC is a universal complexification for G in the category of all complex Lie groups with complex analytic exponential functions. Here, the final assertion is based on a general existence criterion (Theorem 5.4): 0.4.