- #1
Kreizhn
- 743
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Homework Statement
Let G be a Lie group and L be its Lie algebra. Further assume that TG is equiped with a metric tensor g; that is, a (0,2)-covariant tensor that is symmetric and positive-definite on fibres of TG. Assume that [itex] \{ H_i\}_{i=1}^m [/itex] are an orthonormal basis for for L. If [itex] R_X: G \to G [/itex] is the right-translation operator acting as [itex] R_X(Y) = Y\cdot X [/itex] with [itex] \cdot [/itex] the group operation, show that for any [itex]X \in G [/itex] the set [itex] \{ dR_X H_i \}_{i=1}^m [/itex] form an orthonormal basis for [itex] T_X G[/itex].
The Attempt at a Solution
So via recent posts here, we know that [itex] dR_X H_i = H_i X [/itex]. Consider the "inner-product" on [itex] T_X G [/itex] given by [itex] \langle \cdot,\cdot \rangle[/itex]. Then we want to show that
[tex] \langle H_i X, H_j X \rangle = \delta_{ij} [/tex]
Now, I really don't think we can do anything more here without an explicit expression for the inner-product, or at least some consideration of what the group G is.
For my purposes, G is the unitary group [itex] G = \mathfrak U(N) [/itex]. Now normally when we do inner-products on Hilbert spaces, we can normally jump between arguments of the inner-product like
[tex] \langle H_i X, H_j X \rangle = \langle H_i X X^\dagger, H_j \rangle = \langle H_i, H_j \rangle [/tex]
However, I'm really concerned about doing this, for two reasons. The first is that normally, the notion of unitarity requires that we work the inner product defined as an operator on a space and its dual. In this case, the inner-product is on two copies of [itex] T_X G [/itex]. Since the metric is everywhere defined, I suppose we could say that G is a Riemannian manifold, in which case the musical isomorphisms give us a way of relating [itex] T_X^*G [/itex] to [itex] T_XG [/itex] and then it would be fine.
The second reason is "what is unitarity when we change the inner product?" In particular, we know what unitarity is for finite dimensions and [itex] L^2 [/itex], but what does it mean on a Lie algebra? Does it make sense to have unitary operators on the space of skew-Hermitian matrices?