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Mathematics of Computation

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The real cohomology of compact disconnected Lie groups

Author: Robert F. Brown
Journal: Math. Comp. 37 (1973), 255-259
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Abstract: Let G be a compact Lie group with identity component $ {G_0}$ and component group $ \Gamma = G/{G_0}$. The homomorphism $ \chi :G \to {\operatorname{Aut(}}{G_0})$ defined by $ \chi (g)(x) = {g^{ - 1}}xg$ induces $ \chi :\Gamma \to {\operatorname{Aut}}(G)/{\operatorname{Int}}(G)$. The problem of computing the real cohomology $ {H^\ast}(G)$ is solved in the sense that, given $ \chi $, the decomposition of $ \mathfrak{G}$--the Lie algebra of $ {G_0}$, and a description of $ d\chi {(\gamma )_e} \in {\operatorname{Aut}}(\mathfrak{G})$, for each $ \gamma \in \Gamma $, with respect to that decomposition, one can write down a complete description of $ {H^\ast}(G)$ as a Hopf algebra.

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Keywords: Real cohomology, disconnected Lie groups, Hopf algebra, automorphism, simple Lie algebra
Article copyright: © Copyright 1973 American Mathematical Society