Buildings as inner ideal geometries

Dillon, Meighan I.

doi:10.1090/S0002-9939-1995-1264806-X

Buildings as inner ideal geometries
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by Meighan I. Dillon PDF

Proc. Amer. Math. Soc. 123 (1995), 647-650 Request permission

Abstract:

We consider a Chevalley-type group $G(V)$ associated to an integrable representation of a Kac-Moody algebra and show that the associated Kac-Moody group $G(A)$ is a universal cover for $G(V)$. This observation strengthens a result of Kac-Peterson on representations of $G(A)$. It also implies that the building associated to an affine Lie algebra can be realized as an inner ideal geometry.

References

N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123, DOI 10.1007/978-1-4612-1019-1
Meighan I. Dillon, Geometry and affine Lie algebras, J. Algebra 135 (1990), no. 1, 96–111. MR 1076079, DOI 10.1016/0021-8693(90)90151-D
Victor G. Kac, Infinite-dimensional Lie algebras, Progress in Mathematics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 1983. An introduction. MR 739850, DOI 10.1007/978-1-4757-1382-4
V. G. Kac and D. H. Peterson, Defining relations of certain infinite-dimensional groups, Astérisque Numéro Hors Série (1985), 165–208. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837201
Victor G. Kac and Dale H. Peterson, On geometric invariant theory for infinite-dimensional groups, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 109–142. MR 911137, DOI 10.1007/BFb0079235
Victor G. Kac and Dale H. Peterson, Regular functions on certain infinite-dimensional groups, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 141–166. MR 717610
Dale H. Peterson and Victor G. Kac, Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 6, i, 1778–1782. MR 699439, DOI 10.1073/pnas.80.6.1778
Mark Ronan, Lectures on buildings, Perspectives in Mathematics, vol. 7, Academic Press, Inc., Boston, MA, 1989. MR 1005533

Lecture notes on Chevalley groups