Abstract
Everywhere in this section we assume by default that \(\mathbb{k}\) is a field of characteristic zero.
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Notes
- 1.
Also known as \(\mathfrak{g}\)-linear operators and homomorphisms of \(\mathfrak{g}\)-modules (or just \(\mathfrak{g}\)-homomorphisms for short).
- 2.
See Theorem 13.4 on p. 303.
- 3.
Compare with Sect. 6.3 on p. 141.
- 4.
See formula (8.4) on p. 176.
- 5.
Compare with Exercise 8.10 and the preceding paragraph.
- 6.
See formula (16.1) in Sect. 16.1.1 of Algebra I.
- 7.
Meaning that ω ∗ = −ω, where \(\omega ^{{\ast}}: V _{3}^{{\ast}{\ast}}\simeq V _{3}\stackrel{\sim }{\rightarrow }V _{3}^{{\ast}}\) is the dual correlation.
- 8.
See formula (8.9) on p. 180.
- 9.
See Sect. 16.1.1 of Algebra I, especially formula (16.1), and also Example 7.7.
- 10.
Compare with Example 8.3 on p. 179.
- 11.
See Example 11.6 of Algebra I.
- 12.
See Example 11.6 of Algebra I.
- 13.
By definition, the osculating plane to a parametrically given projective curve t ↦ φ(t) at a point φ(a) is spanned by φ(a), φ ′(a) (the velocity), and φ ′ ′(a) (the acceleration) considered as points of the projective space in which the curve lives.
- 14.
Considered as a point of the dual space \(\mathbb{P}_{3}^{\times } = \mathbb{P}(V _{3}^{{\ast}})\).
- 15.
Compare with Problem 8.9.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). \(\mathfrak{s}\mathfrak{l}_{2}\)-Modules. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_8
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