# Questions related to the definition of action of algebraic group $G$ that is locally algebraic

I am just learning about action of $$G$$, algebraic group over an algebraically closed field $$k$$, that is locally algebraic. It states that if $$G$$ acts linearly on a vector space $$W$$, we say the action is locally algebraic if it is locally fintie and for any finite dimensional $$G$$-stable subspace $$V$$, the action $$\theta: G \times V \rightarrow V$$ is a morphism.

1) My first question is what is meant by " $$G$$ acts linearly on $$W$$"?

2) What is meant by " the action $$\theta: G \times V \rightarrow V$$ is a morphism"? Does this mean a morphism as algebraic groups or varieties?

3) Could someone explain why if $$G\times V \rightarrow V$$ is given by $$(g, \sum_{i=1}^n \lambda_i e_i) \rightarrow \sum_{i} \lambda_i h_i(g^{-1})e_i$$ where $$\lambda_i \in k$$, $$\{e_i\}$$ is a basis of $$V$$ and $$h_i \in k[G]$$ then this defines a morphism?

• Can you explain the exact definition of this "locally finite"? – Amira Lo Mar 15 at 13:27

1) $$G$$ acts linearly on $$W$$ means that each element of $$G$$ acts by a linear transformation of $$W$$. Equivalently the action determines a homomorphism of $$G$$ into $$GL(W)$$.
2) This means a morphism of varieties, the algebraic group structure of $$V$$ isn't relevant here.
3) At the level of (closed) points, a morphism of varieties is a map that's locally given by polynomials in affine charts. In this case, and $$h_i(g^{-1})$$ are polynomial in $$g$$, since inversion is a morphism on $$G$$, and $$h_i\in k[G]$$, so this map is polynomial in any affine chart of $$G\times V$$.
• Is an inversion from $G$ to $G$ a polynomial map? – Johnny T. Jun 2 '19 at 21:20
• When you say it's locally given by polynomials, I guess you mean like $f/g$ where $f$ and $g$ are polynomials? (just thinking of $2$ by $2$ matrices the inversion involves $1/$determinant..) – Johnny T. Jun 3 '19 at 7:52
• When I say locally polynomial, I mean that it's a morphism of varieties, equivalently that it's polynomial on affine charts. For $gl(n)$ this ends up being the same thing as being globally given by rational functions whose denominator doesn't vanish, since $gl(n)$ is quasi-affine. If you want to make more sense of all these notions (and properly learn about algebraic groups), you really ought to learn what a variety is, and what morphisms between them are. – A Nonny Mouse Jun 3 '19 at 21:26
• Could you possibly give me an example of what you mean for the case of $GL_2$ by any chance? (where the inverse is a polynomial map) – Johnny T. Jun 3 '19 at 21:50