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I have a small problem that I would appreciate some assistance with. I'm working with the Lie group [itex] \mathfrak U(N) [/itex] of unitary matrices and I have an element [itex] P \in \mathfrak U(N) [/itex]. Furthermore, I have a set

[tex] \left\{ H_1, \ldots, H_d \right\} [/tex]

that generate the Lie algebra [itex] \mathfrak u(N) [/itex] of skew-Hermitian matrices under a Lie bracket given by a commutator [itex] [A,B] = AB-BA [/itex].

My overall goal is to express P as an exponential product

[tex] P = \exp\left( \alpha_1 H_1 \right) \cdots \exp\left( \alpha_d H_d \right) [/tex]

for [itex] \alpha_i \in \mathbb R [/itex]. I know in general that this decomposition is probably not a simple one. On the other hand, I do know that I can express P as

[tex] P = \exp\left( - \mu G \right) [/tex]

for some [itex] G \in \mathfrak u(N) [/itex]. It seems to me then that maybe I can reduce this to simply finding a decomposition of G in terms of the generating set of the Lie algebra. Even if the decomposition is in terms of the Lie bracket, I should be able to exploit properties of the matrix exponential to get what I desire.

In any case, I am not certain if there is an easier way of doing this. Or if we can indeed just focus on decomposing G in terms of [itex] H_i [/itex], how would I do this?

P.S.If it helps, I can instead choose to work in the special unitary group [itex] \mathfrak{SU}(N) [/itex] and hence [itex] \mathfrak{su}(N) [/itex] the Lie algebra of traceless skew-Hermitian matrices. Also, I can write

[tex] P = \exp\left[ \alpha_1 \sum_{i \in I_1} H_i \right] \cdots \exp\left[\alpha_k \sum_{i \in I_k} H_i \right] [/tex]

For [itex] I_j \subseteq \left\{1, \ldots, d \right\} [/itex]. That is, the argument of the exponentials need not be only a single generating element, but it can also be a sum of generators.

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# Lie algebraic decomposition

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