Podemos definir una representación de un grupo de Lie y obtener la representación inducida del álgebra de Lie. Sea $G$ actuar $V$ y $W$ , $\mathfrak{g}$ sea el álgebra de Lie asociada a $G$ y $X \in \mathfrak{g}$ . Entonces la acción sobre $V$ se define como $\displaystyle X(v)=\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)$ donde $\gamma_t$ es un camino en $G$ con $\gamma'_0 = X$ . Entonces: $$ \begin{align*} X(v \otimes w) & = \left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\otimes \gamma_t(w) \\ & \stackrel{?}{=}\left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\right)\otimes w + v\otimes \left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\right) \\ & = X(v)\otimes w + v \otimes x(w)\end{align*}$$ Mi pregunta es: ¿por qué se divide?
Respuesta
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$$ \begin{align*} X(v \otimes w) & = \left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\otimes \gamma_t(w) \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes \gamma_t(w)-v \otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes ( \gamma_t(w)-w+w)-v \otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes (\gamma_t(w)-w)+ \gamma_t(v)\otimes w-v \otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes (\gamma_t(w)-w)+ (\gamma_t(v)-v)\otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{v \otimes (\gamma_t(w)-w)}{t}+ \lim_{t \to 0} \,\,\, \frac{(\gamma_t(v)-v)\otimes w}{t} \\ & =\left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\right)\otimes w + v\otimes \left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(w)\right) \\ & = X(v)\otimes w + v \otimes X(w)\end{align*}$$
