Abstract
Suppose \(\mathfrak{g}\) is a complex reductive Lie algebra. The invariant subspace of the exterior algebra \(\wedge(\mathfrak{g})\) is a graded super Hopf algebra; let \(P(\mathfrak{g})\subset \wedge(\mathfrak{g})^{\mathfrak{g}}\) be the graded subspace of primitive elements. The central results in this chapter are the Hopf–Koszul–Samelson Theorem identifying \((\wedge\mathfrak{g})^{\mathfrak{g}}\) with the exterior algebra over \(P(\mathfrak{g})\), and the Transgression Theorem, showing that the space of primitive elements coincides with the image of the transgression map for the Weil algebra. The proofs make extensive use of Lie algebra homology and cohomology.
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References
H. Cartan. La transgression dans un groupe de Lie et dans un fibré principal. In Colloque de topologie (espaces fibrés) (Bruxelles), pages 73–81, 1950, centre belge de recherches mathématiques, Georges Thone, Liège, Masson et Cie., Paris.
V. Chevalley. The Betti numbers of the exceptional groups. In Proceedings of the International Congress of Mathematicians, Volume 2, pages 21–24, 1950.
W. Greub, S. Halperin, and R. Vanstone. Connections, Curvature, and Cohomology. Academic Press, New York, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces.
B. Kostant. Clifford algebra analogue of the Hopf–Koszul–Samelson theorem, the ρ-decomposition \(\mathrm{C}(\mathfrak{g})=\mathrm {End}\,{V}\sb{\rho}\otimes \mathrm{C}(\mathrm{P})\), and the \(\mathfrak{g}\)-module structure of \(\bigwedge\mathfrak{g}\). Adv. Math., 125(2):275–350, 1997.
J.-L. Koszul. Homologie et cohomologie des algèbres de Lie. Bull. Soc. Math. Fr., 78:65–127, 1950.
J. Leray. Sur l’homologie des groupes de Lie, des espaces homogènes et des espaces fibrés principaux. In Colloque de topologie (espaces fibrés) (Bruxelles), pages 101–115, 1950. Centre belge de recherches mathématiques, Georges Thone, Liège, Masson et Cie., Paris.
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Meinrenken, E. (2013). The Hopf–Koszul–Samelson Theorem. In: Clifford Algebras and Lie Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36216-3_10
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DOI: https://doi.org/10.1007/978-3-642-36216-3_10
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