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Barcelona Algebraic Topology Group

University: | Universitat Autònoma de Barcelona | ||
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Status: | Finished | Degree: | Phd |

Directors: | Student: | ||

Date: | 02-07-2001 | ||

Digital version | |||

Description: | |||

The study of the maps between classifying spaces of compact Lie groups has been one of the highlights of algebraic topology in the final quarter of the XXth century. a=2, _{i,i}a ≥ 0 and _{i,j}a=0 implies _{i,j}a=0._{j,i}All this proces can be inverted and we can recover the Lie algebra from the Cartan matrix and ``integrate'' this Lie algebra to obtain a simply connected, compact, connected Lie group. Consider now a generalizad Cartan matrix, that is a non necessarily positive definite Cartan matrix. We can construct an integrable Lie algebra (non finite dimensional, in general) and from it a topologicla group. The result of these constructions are the called Kac-Moody algebras and Kac-Moody groups. From a homotopy point of view the Kac-Moody groups were studied by N. Kitchloo (cohomological properties) and this results took us to consider other well known results in compact Lie groups to be generalized to Kac-Moody groups. The main result of the thesis is the study of the mapping space [ BK,BK], where K is a rank 2 Kac-Moody group.In order to understand [ BK,BK] we must calculate [BT,BK], where T is a maximal torus in K. Here we get results which do not agree with the compact Lie group case: there exist maps from BT to BK which do not come from representations.With this study we get a complete description of the subspace of [ BT,BK] which involve all the maps which come from maps of [BK,BK]. This classification will allow us to understand the space [BK,BK], after proving that the map induced by the inclusion [BK,BK] in [BT,BK] is injective.Studying this we get other results like the characterization of the homotopy type of the rank 2 Kac-Moody groups (in particular we obtain non-isomorphic Kac-Moody groups with the same classifying space) and a characterization of the possible degrees of maps from BK to BK. |