Given the matrix $S ∈ M_n (\Bbb C)$ then the following forms a lie group $$G= \{ a ∈ M_n (\Bbb C)| aSa^† =S \}$$ I'm completely lost on how to find the Lie algebra of this group.
Finding Lie algebra of Lie group
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group-theory
lie-groups
lie-algebras
1 Answers
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My suggestion is to pick a path $\gamma: (-\epsilon, \epsilon) \to G$ such that $\gamma(0) = I$ and $\gamma'(0) = A$ for some matrix $A$. Observe that
$$ \gamma(t)S \gamma^\dagger(t) \;\; =\;\; S. $$
Take the derivative of both sides and evaluate at $t=0$. See what constraints you get for $A$.