Representation of a Lie algebra over the space of smooth functions.
$begingroup$
Suppose that $mathfrak{g}$ is the Lie algebra of a compact and connected linear Lie group. Now let's consider the mapping
$$rho : mathfrak{g} longrightarrow
mathfrak{gl}({mathcal{C}^{infty}(G, mathbb{C})} )$$
defined by
$$rho(X)f(x) = frac{d}{dt}Bigr|_{t=0}f(xexp(tX))$$
for all
$$Xinmathfrak{g},,, finmathcal{C}^{infty}(G, mathbb{C})
,,,, mbox{and} ,,,, xin G$$
In the book, "Unitary Representations and Harmonic Analysis" by Mitsuo Sugiura, page 75, it is stated without proof, that the map $rho$ is a Lie algebra representation. Can anyone give me any suggestions to prove that the $rho$ map is linear?
functional-analysis differential-geometry lie-groups lie-algebras smooth-manifolds
$endgroup$
add a comment |
$begingroup$
Suppose that $mathfrak{g}$ is the Lie algebra of a compact and connected linear Lie group. Now let's consider the mapping
$$rho : mathfrak{g} longrightarrow
mathfrak{gl}({mathcal{C}^{infty}(G, mathbb{C})} )$$
defined by
$$rho(X)f(x) = frac{d}{dt}Bigr|_{t=0}f(xexp(tX))$$
for all
$$Xinmathfrak{g},,, finmathcal{C}^{infty}(G, mathbb{C})
,,,, mbox{and} ,,,, xin G$$
In the book, "Unitary Representations and Harmonic Analysis" by Mitsuo Sugiura, page 75, it is stated without proof, that the map $rho$ is a Lie algebra representation. Can anyone give me any suggestions to prove that the $rho$ map is linear?
functional-analysis differential-geometry lie-groups lie-algebras smooth-manifolds
$endgroup$
add a comment |
$begingroup$
Suppose that $mathfrak{g}$ is the Lie algebra of a compact and connected linear Lie group. Now let's consider the mapping
$$rho : mathfrak{g} longrightarrow
mathfrak{gl}({mathcal{C}^{infty}(G, mathbb{C})} )$$
defined by
$$rho(X)f(x) = frac{d}{dt}Bigr|_{t=0}f(xexp(tX))$$
for all
$$Xinmathfrak{g},,, finmathcal{C}^{infty}(G, mathbb{C})
,,,, mbox{and} ,,,, xin G$$
In the book, "Unitary Representations and Harmonic Analysis" by Mitsuo Sugiura, page 75, it is stated without proof, that the map $rho$ is a Lie algebra representation. Can anyone give me any suggestions to prove that the $rho$ map is linear?
functional-analysis differential-geometry lie-groups lie-algebras smooth-manifolds
$endgroup$
Suppose that $mathfrak{g}$ is the Lie algebra of a compact and connected linear Lie group. Now let's consider the mapping
$$rho : mathfrak{g} longrightarrow
mathfrak{gl}({mathcal{C}^{infty}(G, mathbb{C})} )$$
defined by
$$rho(X)f(x) = frac{d}{dt}Bigr|_{t=0}f(xexp(tX))$$
for all
$$Xinmathfrak{g},,, finmathcal{C}^{infty}(G, mathbb{C})
,,,, mbox{and} ,,,, xin G$$
In the book, "Unitary Representations and Harmonic Analysis" by Mitsuo Sugiura, page 75, it is stated without proof, that the map $rho$ is a Lie algebra representation. Can anyone give me any suggestions to prove that the $rho$ map is linear?
functional-analysis differential-geometry lie-groups lie-algebras smooth-manifolds
functional-analysis differential-geometry lie-groups lie-algebras smooth-manifolds
asked Jan 27 at 21:04


Pedro ZapataPedro Zapata
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