Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation : $$\Gamma : Lie(G) \to gl(V)$$

If yes, how ? Please help me understand this. I would appreciate if the explanation is simple and from all prespectives like thinking of $Lie(G)$ as derivations on the coordinate ring or as the tangent space at identity of $G$.

  • $\begingroup$ What is the characteristic of $G$? $\endgroup$ – Tsemo Aristide Feb 3 '16 at 16:59
  • $\begingroup$ @TsemoAristide It is over a field of characterstic zero. I have edited that in the question too. $\endgroup$ – Jagdeep Singh Feb 3 '16 at 17:18
  • $\begingroup$ Take the differential of $\rho\colon G\rightarrow GL(V)$, see here. $\endgroup$ – Dietrich Burde Feb 4 '16 at 7:54

A linear Group is also a linear map. Clearly linear maps can be represented as matrices. For a representation in $gl(V)$ it is required that no Matrix has vanishing determinant. This ensures in the case that $K$ has characteristic Zero that the linear Group is invertible. And Groups must be invertible per definition.

Therefore this representation exists.


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