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Kindly asking for any hints about the following questions:

Assume $g$ is a finite-dimensional Lie algebra. We denote the group of Lie algebra automorphisms of $g$ by $\rm Aut_k g$. Any Lie algebra automorphism of $g$, can also be viewed as an automorphism of $g$ considered as a scheme. Why?

Thanks for your help!

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Simply because every linear automorphism o a vector space $V$ is an automorphism of $V$ viewed as a scheme. That $g$ is a Lie algebra or that the map is an automorphism of Lie algebras has nothing to do with this. –  Mariano Suárez-Alvarez Jan 29 '13 at 22:41
    
Why, For every finite dimensional vector space $V$, we can viewed $V$ as an scheme? –  Masoud Jan 29 '13 at 22:45
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@Masoud, let $V$ be an $n$-dimensional vector space over $k,$ then $V \cong k^n$ and $k^n=\mathbb{A}_k^n=\text{Spec}(k[x_1,\cdots,x_n])$ right? –  Ehsan M. Kermani Jan 30 '13 at 7:13
    
that's right, Thank you. Now, Do any morphism of schemes be a continuous map between schemes? –  Masoud Jan 30 '13 at 12:01
    
Dear @Masoud, a scheme is a locally ringed space and a morphism between two locally ringed spaces is a continuous map between their underlying topology satisfying two more conditions. You may look through the relevant definitions of schemes in Hartshorne's algebraic geometry or Ravi Vakil's notes. –  Ehsan M. Kermani Jan 30 '13 at 19:35
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