Asian Journal of Mathematics

Kac-Moody groups, infinite dimensional differential geometry and cities

Walter Freyn

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Minimal affine Kac-Moody groups act on affine twin buildings by isometries. However there is no way to extend this action to any completion of the Kac-Moody groups. To remedy this, we introduce in this paper affine twin cities, a new class of objects, whose elements behave like completions of twin buildings. Twin cities are defined as special arrays of affine buildings connected among themselves by twinnings. Corresponding to completed affine Kac-Moody groups they are characterized by the type of the affine buildings and by some kind of "regularity conditions" describing the completion. The isometry groups of affine twin cities are (completions of) affine Kac-Moody groups. We study applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical differential geometric results: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In two sequels we will describe the theory of twin cities for formal completions.

Article information

Asian J. Math., Volume 16, Number 4 (2012), 607-636.

First available in Project Euclid: 12 December 2012

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Zentralblatt MATH identifier

Primary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 20G44: Kac-Moody groups 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05] 58B99: None of the above, but in this section

Loop group affine Kac-Moody group twin city twin building polar action Kac-Moody geometry


Freyn, Walter. Kac-Moody groups, infinite dimensional differential geometry and cities. Asian J. Math. 16 (2012), no. 4, 607--636.

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