##
**Banach-Lie quotients, enlargibility, and universal complexifications.**
*(English)*
Zbl 1029.22029

The authors study various questions about the existence of Banach-Lie groups with prescribed Lie algebras. Their first main result asserts that the quotient of a Banach-Lie group by a Lie subgroup always carries the structure of a Banach-Lie group. So far one had to make extra hypotheses on the subgroup to obtain this conclusion. This result they use to reprove a characterization (due to van Est) of enlargible Lie algebras, i.e. those Banach-Lie algebras which occur as Lie algebras of Banach-Lie groups.

There are two kinds of universal objects considered for Banach-Lie algebras: a universal “enlargible envelope”, which is a Banach-Lie algebra through which all homomorphisms into enlargible Lie algebras factor, and a universal complexification which is defined in analogy to the finite-dimensional case. In general, neither of these objects needs to exist. The authors give necessary and sufficient conditions for the existence of both.

There are two kinds of universal objects considered for Banach-Lie algebras: a universal “enlargible envelope”, which is a Banach-Lie algebra through which all homomorphisms into enlargible Lie algebras factor, and a universal complexification which is defined in analogy to the finite-dimensional case. In general, neither of these objects needs to exist. The authors give necessary and sufficient conditions for the existence of both.

Reviewer: Joachim Hilgert (Clausthal)

### MSC:

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

22E15 | General properties and structure of real Lie groups |

22E10 | General properties and structure of complex Lie groups |

PDFBibTeX
XMLCite

\textit{H. Glöckner} and \textit{K.-H. Neeb}, J. Reine Angew. Math. 560, 1--28 (2003; Zbl 1029.22029)

Full Text:
DOI

### References:

[1] | Bo N., Chapter pp 1– (1989) |

[2] | Douady M., Invent. Math. 1 pp 133– (1966) |

[3] | Dupre J. F, J. Lie Th. 10 pp 359– (2000) |

[4] | Es W. T, Wet. 65 pp 391– (1962) |

[5] | Wet. 67 pp 15– (1964) |

[6] | [Gl1] H. Gl ckner, Infinite-Dimensional Complex Groups and Semigroups: Representations of Cones, Tubes, and Conelike Semigroups, Doctoral Dissertation, TU Darmstadt, February 2000. |

[7] | Gl H, J. Funct. Anal. 194 pp 347– (2002) |

[8] | [Ho1] G. Hochschild, The Structure of Lie Groups, Holden-Day, 1965. |

[9] | Ho G, Trans. Amer. Math. Soc. 125 pp 406– (1966) |

[10] | Le L, Contemp. Math. 205 pp 169– (1997) |

[11] | [Ma] P. Maier, Central extensions of topological current algebras, in: Geometry and Analysis on Finite and Infinite-Dimensional Lie Groups, Strasburger, A. et al., eds., Banach Center Publ. 55, Warsaw, 61-76. |

[12] | Ms B., Acta Math. 108 pp 229– (1962) |

[13] | Mi E, Can. J. Math. 11 pp 556– (1959) |

[14] | [Ne1] K.H. Neeb, Infinite-dimensional Lie groups and their representations, Lectures at the European School in Group Theory, SDU-Odense Univ., August 2000 (available also as Preprint 2206 of TU Darmstadt, April 2002). |

[15] | Ne K., Ann. Inst. Fourier Grenoble 52 pp 1365– (2002) |

[16] | [Ne3] K.H. Neeb, Borel-Weil theory for loop groups, in: Infinite Dimensional K hler Manifolds, Huckleberry, A. and T. Wurzbacher, eds., DMV-Seminar 31, Birkh user Verlag (2001), 179-229. · Zbl 0994.22014 |

[17] | [Ne4] K.H. Neeb, Classical Hilbert-Lie groups, their extensions and their homotopy groups, in: Geometry and Analysis on Finite and Infinite-Dimensional Lie Groups, Strasburger, A. et al., eds., Banach Center Publ. 55, Warsaw, 87-151. |

[18] | Ne K., Geom. Dedic. 95 pp 115– (2002) |

[19] | Pe V. G, Nova J. Alg. Geom. 1 pp 371– (1992) |

[20] | Pe V. G, Bull. Aust. Math. Soc. 48 pp 13– (1993) |

[21] | Sw S, Trans. Amer. Math. Soc. 128 pp 291– (1967) |

[22] | Sw S., Wet. 74 pp 235– (1971) |

[23] | [Te] J. Teichmann, Infinite Dimensional LieGroups with a View Towards Functional Analysis, Ph.D. thesis, University of Vienna, 1999. |

[24] | [Up] H. Upmeier, Symmetric Banach Manifolds and Jordan C-Algebras, North-Holland, Amsterdam1985. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.