Cohomology of homomorphisms of Lie algebras and Lie groups

Robert F. Brown

Vol. 69 (1977), No. 2, 325–332

DOI: 10.2140/pjm.1977.69.325

Abstract

Given compact, connected Lie groups G₁ and G₂ and given h : G₁ → G₂ a homomorphism with kernel K, let Ph^∗ : PH^∗(G₂) → PH^∗(G₁) be the homomorphism of the primitives in the real cohomology induced by h. We prove that if the rank of G₂ is greater than or equal to the rank of G₁, then the dimension of the kernel of Ph^∗ is greater than or equal to the rank of K. We discuss when the inequality is an equality and we use the inequality to study when the hypothesis that Ph^∗ is an isomorphism implies that h itself is an isomorphism.

Mathematical Subject Classification

Primary: 57F10, 57F10

Milestones

Received: 25 October 1976

Published: 1 April 1977