I've started Lie algebras course this semester, so I already have some questions. Well, I need to find a $Lie(SL_n(\mathbb{K}))$ (Lie algebra). Field $\mathbb{K}$ is algebraically closed and $char( \mathbb{K})=0$. Teacher gave me next piece of advice. This space is given by the equation $detg=1,\;g\in GL_n$. We have a formula for determinant of matrix $g$.
$$detg=\sum_{\sigma \in S_n}sgn(\sigma)a_{1\sigma (1)}...a_{n\sigma (n)}$$
So I need to differentiate this identity at E. But I thought that $a_{ij}$ are elements of out field, so $d(a_{ij})=0$. Anyway I know, that I need to get something like this:
$$d(detg)=\sum_{i=1}^nda_{ii}=0$$
So the Lie algebra of $SL_n$ is subspace of matrices with zero trace. Please, help me understand the differentiating part.
Thanks.
Well I'll try to answer my question on my own. Let's differentiate first sum. $$ d(detg)=\sum_{\sigma \in S_n} (-1)^\sigma\sum_{i=1}^nd(a_{i\sigma(i)})a_{1\sigma(1)}...\hat{a_{i\sigma(i)}}...a_{n\sigma(n)}=0$$ Now lets restrict it to the E. Members of this sum are not zero iff $\sigma=id$.
$$d(detg)\vert_E=\sum_{i=1}^nd(a_{ii})=0$$