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The enveloping group of a Lie algebra. (English) Zbl 0684.22010
Proc. Winter Sch. Geom. Phys., Srní/Czech. 1988, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 355-367 (1989).
[For the entire collection see Zbl 0672.00006.]
The local structure of a finite dimensional Lie group is determined by the structure of the corresponding Lie algebra through the exponential map. For the infinite dimensional Lie groups, this connection in general fails. In this paper, the author introduces the concept of polynomial groups of a topological group and of a Lie algebra and establishes the isomorphism of these two objects and potentially gives an analytic description of a group in terms of its Lie algebra. Let G be a topological group, C(R,G) be the topological group of all continuous G- valued functions on the real line R and \(\Lambda\) (G) be the set of all continuous homomorphisms of the additive group of reals into G (i.e. one parameter subgroups of G). The polynomial group P(G) of G is defined to be the subgroup of C(R,G) generated by \(\Lambda\) (G). Let \({\mathfrak g}\) be a Lie algebra and \(T=\coprod_{n}T_ n\) be the tensor algebra of the linear space \({\mathfrak g}\) and \(\hat T=\prod_{n}T_ n\) be the Magnus algebra, where \(T_ n\) is the homogeneous component of order n of T; \({\mathcal L}({\mathfrak g})\) be the R-group of all G-valued functions on R which are finite pointwise products of exponential functions. For \(f\in {\mathcal L}({\mathfrak g})\), set log f(t)\(=\sum_{n}p_ n(f)t^ n\), \(q_ n=j\circ p_ n\), where j is the homomorphism of the Lie subalgebra L of T generated by \({\mathfrak g}=T_ 1\) into \({\mathfrak g}\) and \(Q=\cap_{n}Ker q_ n\). The polynomial group P(\({\mathfrak g})\) of \({\mathfrak g}\) is defined to be \({\mathcal L}({\mathfrak g})/Q\). It is shown that P(\({\mathfrak g})\) is the unique R-Lie group associated with \({\mathfrak g}\). In particular, for a Banach Lie group G with the Lie algebra \({\mathfrak g}\), the R-Lie groups P(G) and P(\({\mathfrak g})\) are isomorphic.
Reviewer: E.Abe
MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E60 Lie algebras of Lie groups
17B45 Lie algebras of linear algebraic groups