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Claudio Gorodski
Claudio Gorodski
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For a left-invariant vector field $X$ on $G$, denote by $X^R$ the right-invariant vector field with the same value at the identity. From $\iota_*X=\iota_*L_{g*}X=R_{g^{-1}*}\iota_*X$, we see that $\iota_*X$ is right-invariant, where $\iota$ is the inversion map $g\mapsto g^{-1}$. Since $\iota_*X(1)=-X(1)$, we get $X^R=-\iota_*X$. Finally, $[X^R,Y^R]=[-\iota_*X, -\iota_*Y]=\iota_*[X,Y]=-[X,Y]^R$$[X^R,Y^R]=[-\iota_*X,-\iota_*Y]=\iota_*[X,Y]=-[X,Y]^R$.

For a left-invariant vector field $X$ on $G$, denote by $X^R$ the right-invariant vector field with the same value at the identity. From $\iota_*X=\iota_*L_{g*}X=R_{g^{-1}*}\iota_*X$, we see that $\iota_*X$ is right-invariant, where $\iota$ is the inversion map $g\mapsto g^{-1}$. Since $\iota_*X(1)=-X(1)$, we get $X^R=-\iota_*X$. Finally, $[X^R,Y^R]=[-\iota_*X, -\iota_*Y]=\iota_*[X,Y]=-[X,Y]^R$.

For a left-invariant vector field $X$ on $G$, denote by $X^R$ the right-invariant vector field with the same value at the identity. From $\iota_*X=\iota_*L_{g*}X=R_{g^{-1}*}\iota_*X$, we see that $\iota_*X$ is right-invariant, where $\iota$ is the inversion map $g\mapsto g^{-1}$. Since $\iota_*X(1)=-X(1)$, we get $X^R=-\iota_*X$. Finally, $[X^R,Y^R]=[-\iota_*X,-\iota_*Y]=\iota_*[X,Y]=-[X,Y]^R$.

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Claudio Gorodski
Claudio Gorodski
  • 4.7k
  • 1
  • 28
  • 44

For a left-invariant vector field $X$ on $G$, denote by $X^R$ the right-invariant vector field with the same value at the identity. From $\iota_*X=\iota_*L_{g*}X=R_{g^{-1}*}\iota_*X$, we see that $\iota_*X$ is right-invariant, where $\iota$ is the inversion map $g\mapsto g^{-1}$. Since $\iota_*X(1)=-X(1)$, we get $X^R=-\iota_*X$. Finally, $[X^R,Y^R]=[-\iota_*X, -\iota_*Y]=\iota_*[X,Y]=-[X,Y]^R$.