Abstract
The representation of Lie groups is closely related to the representation of their Lie algebras, and we shall discuss them later in this chapter. In the case of compact groups, however, there is a well developed representation theory, which we shall consider in the first section. Before discussing compact groups, let us state a definition and a proposition that hold for all Lie groups.
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Notes
- 1.
This can always be done by rescaling the volume element.
- 2.
When a symmetric group is considered as an abstract group—as opposed to a group of transformations—we may multiply permutations (keep track of how each number is repeatedly transformed) from left to right. However, since the permutations here act on vectors on their right, it is more natural to calculate their products from right to left.
- 3.
We use the word “local” to mean the collection of all points that can be connected to the identity by a curve in the Lie group G. If this collection exhausts G, then we say that G is connected. If, furthermore, all closed curves (loops) in G can be shrunk to a point, we say that G is simply connected. The word “local” can be replaced by “simply connected” in what follows.
- 4.
Recall that the rank of \(\mathfrak{g}\) is the dimension of the Cartan subalgebra of \(\mathfrak{g}\).
- 5.
Sometimes we use
, 
, and 
instead of 
, 
, and 
. - 6.
Please make sure to differentiate between the pair (M ij ,P k ) (which acts on p) and the pair
, which acts on the state vectors in the Hilbert space of representation. - 7.
This “angular momentum” includes ordinary rotations as well as the Lorentz boosts.
- 8.
The reader should be warned that although such a rotation does not change p, the rotation operator may change the state |ψ p 〉. However, the resulting state will be an eigenstate of the
’s with eigenvalue p. - 9.
We are using the fact that O(p,n−p) is transitive (see Problem 30.15).
- 10.
We use units in which c=1.
, 
, and 
instead of 
, 
, and 
.
, which acts on the state vectors in the Hilbert space of representation.
’s with eigenvalue p.References
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Mackey, G.: Induced Representations. Benjamin, Elmsford (1968)
Miller, W.: Lie Theory and Special Functions. Academic Press, New York (1968)
Varadarajan, V.: Lie Groups, Lie Algebras and Their Representations. Springer, Berlin (1984)
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Hassani, S. (2013). Representation of Lie Groups and Lie Algebras. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_30
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