We have that A and B belong to different representations of the same Lie Group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]A = e^{tX}[/itex]

[itex]B = e^{tY}[/itex]

We want to show, for a specific matrix M

[itex]B^{-1} M B = AM[/itex]

Does it suffice to show this to first order?

[itex]\left(1 -tY + \dots \right)M \left(1 + tY + \dots \right) = \left(1 + tX + \dots \right)M[/itex]

In other words is

[itex]-YM + MY = XM[/itex]

sufficient to show

[itex]B^{-1} M B = AM[/itex]

for all t?

I have seen this used in physics derivations, but it's not clear to me if and why this is sufficient.

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# Equality involving matrix exponentials / Lie group representations

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