Abstract
We discuss two classical results in homological algebra of modules over an enveloping algebra – lemmas of Casselman–Osborne and Wigner. They have a common theme: they are statements about derived functors. While the statements for the functors itself are obvious, the statements for derived functors are not and the published proofs were completely different from each other. First we give simple, pedestrian arguments for both results based on the same principle. Then we give a natural generalization of these results in the setting of derived categories.
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Notes
- 1.
\(\mathcal{A}^{\circ }\) is the category opposite to \(\mathcal{A}\).
- 2.
I do not know any example where this result fails in unbounded case.
- 3.
I would prefer a proof of the next theorem which doesn’t use the construction of the derived functor, but its universal property. Unfortunately, I do not know such argument.
References
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Nicolas Bourbaki, Groupes et algèbres de Lie, Hermann, 1968.
William Casselman and M. Scott Osborne, The \(\mathfrak{n}\) -cohomology of representations with an infinitesimal character, Compositio Math. 31 (1975), no. 2, 219–227.
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996.
Anthony W. Knapp and David A. Vogan, Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, Vol. 45, Princeton University Press, Princeton, NJ, 1995.
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© 2014 Springer International Publishing Switzerland
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Miličić, D. (2014). Variations on a Casselman–Osborne Theme. In: Mason, G., Penkov, I., Wolf, J. (eds) Developments and Retrospectives in Lie Theory. Developments in Mathematics, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-09804-3_13
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DOI: https://doi.org/10.1007/978-3-319-09804-3_13
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