Abstract
The Lie algebra tale continues in this chapter with the essential and fundamental issue of linear representations.
My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful....
Hermann Weyl
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Notes
- 1.
A subspace \(W\subset V\) is named invariant if it is mapped into itself by all group elements.
- 2.
In this context it is quite convenient to utilize the notations and the nomenclature of quantum mechanics where, as basis of the Hilbert space of physical states, one utilizes the eigenstates of a complete set of commuting observable operators \(\mathscr {O}_{1,2,\dots , n}\). According with Dirac, these eigenstates are denoted \(|O_1,O_2, \dots , O_n>\) naming \(O_{1,2,\dots , n}\) the eigenvalues of the considered observables. See later on in this chapter for an introduction to functional spaces and the Hilbert space.
- 3.
Indeed the possible number of \(\alpha \) root subtractions from a given weight \(\mathbf {w}\) is completely determined by the scalar product of the weight with the root \(< \mathbf {w} \, , \,\alpha>\).
- 4.
Look back at Sect. 3.2.5.
- 5.
In the definition below, for simplicity we confine ourselves to the case where the functional space is composed of functions of only one variable x.
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Fré, P.G. (2018). Hermann Weyl and Representation Theory. In: A Conceptual History of Space and Symmetry . Springer, Cham. https://doi.org/10.1007/978-3-319-98023-2_6
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DOI: https://doi.org/10.1007/978-3-319-98023-2_6
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