Reference : Some unexpected facts about Lie Algebras
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Some unexpected facts about Lie Algebras
Leytem, Alain mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Doctorand seminar
22-04-2013 and 29-04-2013
Université du Luxembourg
[en] Lie algebra ; categories ; splitting lemma
[en] Lie algebras and their morphisms do not behave as nicely as one might expect. We will discuss some of the most important illustrations. For example, the category of Lie algebras admits direct products, but no coproducts, and the object called "direct sum" is actually not a direct sum in the categorical sense. It admits kernels and cokernels, but it is neither abelian, nor additive, and the image of a morphism is not a categorical image. Nevertheless, one can still define short exact sequences of Lie algebras. We finish by discussing the Splitting Lemma, which for Lie algebras, in general, only holds in a weaker form than in the abelian case.

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