Representation of algebraic groups
$begingroup$
I have a question related with the notion of representation of algebraic groups. Let $G$ an algebraic group over $k$ and let $V$ a finite dimensional $k$-vector space. We have that $mathbb{V}=rm{Spec}(rm{S}^{bullet}_{k}(V^{ast}))$ is an algebraic variety whose rational points are the points of $V$. I read that a representation of an algebraic group is to give a morphism of algebraic groups
$$rho:Grightarrowrm{Gl}(V)text{,}$$
but I dont understand how I should interpret $rm{Gl}(V)$, I think it should be $rm{Gl}(mathbb{V})$, because to give a representation of a group over a vector space is equivalent to give an action of $G$ over $V$.
Thank you for your time.
representation-theory algebraic-groups
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
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add a comment |
$begingroup$
I have a question related with the notion of representation of algebraic groups. Let $G$ an algebraic group over $k$ and let $V$ a finite dimensional $k$-vector space. We have that $mathbb{V}=rm{Spec}(rm{S}^{bullet}_{k}(V^{ast}))$ is an algebraic variety whose rational points are the points of $V$. I read that a representation of an algebraic group is to give a morphism of algebraic groups
$$rho:Grightarrowrm{Gl}(V)text{,}$$
but I dont understand how I should interpret $rm{Gl}(V)$, I think it should be $rm{Gl}(mathbb{V})$, because to give a representation of a group over a vector space is equivalent to give an action of $G$ over $V$.
Thank you for your time.
representation-theory algebraic-groups
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
$endgroup$
$begingroup$
If we have a representation $Grightarrow rm{Gl}(V)$, do not we have an action of $G$ over $mathbb{V}$?
$endgroup$
– J. Ove
Dec 11 '18 at 16:39
$begingroup$
Sorry, I have just understand it x)
$endgroup$
– J. Ove
Dec 11 '18 at 16:43
add a comment |
$begingroup$
I have a question related with the notion of representation of algebraic groups. Let $G$ an algebraic group over $k$ and let $V$ a finite dimensional $k$-vector space. We have that $mathbb{V}=rm{Spec}(rm{S}^{bullet}_{k}(V^{ast}))$ is an algebraic variety whose rational points are the points of $V$. I read that a representation of an algebraic group is to give a morphism of algebraic groups
$$rho:Grightarrowrm{Gl}(V)text{,}$$
but I dont understand how I should interpret $rm{Gl}(V)$, I think it should be $rm{Gl}(mathbb{V})$, because to give a representation of a group over a vector space is equivalent to give an action of $G$ over $V$.
Thank you for your time.
representation-theory algebraic-groups
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
$endgroup$
I have a question related with the notion of representation of algebraic groups. Let $G$ an algebraic group over $k$ and let $V$ a finite dimensional $k$-vector space. We have that $mathbb{V}=rm{Spec}(rm{S}^{bullet}_{k}(V^{ast}))$ is an algebraic variety whose rational points are the points of $V$. I read that a representation of an algebraic group is to give a morphism of algebraic groups
$$rho:Grightarrowrm{Gl}(V)text{,}$$
but I dont understand how I should interpret $rm{Gl}(V)$, I think it should be $rm{Gl}(mathbb{V})$, because to give a representation of a group over a vector space is equivalent to give an action of $G$ over $V$.
Thank you for your time.
representation-theory algebraic-groups
representation-theory algebraic-groups
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
asked Dec 11 '18 at 16:26
J. OveJ. Ove
111
111
$begingroup$
If we have a representation $Grightarrow rm{Gl}(V)$, do not we have an action of $G$ over $mathbb{V}$?
$endgroup$
– J. Ove
Dec 11 '18 at 16:39
$begingroup$
Sorry, I have just understand it x)
$endgroup$
– J. Ove
Dec 11 '18 at 16:43
add a comment |
$begingroup$
If we have a representation $Grightarrow rm{Gl}(V)$, do not we have an action of $G$ over $mathbb{V}$?
$endgroup$
– J. Ove
Dec 11 '18 at 16:39
$begingroup$
Sorry, I have just understand it x)
$endgroup$
– J. Ove
Dec 11 '18 at 16:43
$begingroup$
If we have a representation $Grightarrow rm{Gl}(V)$, do not we have an action of $G$ over $mathbb{V}$?
$endgroup$
– J. Ove
Dec 11 '18 at 16:39
$begingroup$
If we have a representation $Grightarrow rm{Gl}(V)$, do not we have an action of $G$ over $mathbb{V}$?
$endgroup$
– J. Ove
Dec 11 '18 at 16:39
$begingroup$
Sorry, I have just understand it x)
$endgroup$
– J. Ove
Dec 11 '18 at 16:43
$begingroup$
Sorry, I have just understand it x)
$endgroup$
– J. Ove
Dec 11 '18 at 16:43
add a comment |
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$begingroup$
If we have a representation $Grightarrow rm{Gl}(V)$, do not we have an action of $G$ over $mathbb{V}$?
$endgroup$
– J. Ove
Dec 11 '18 at 16:39
$begingroup$
Sorry, I have just understand it x)
$endgroup$
– J. Ove
Dec 11 '18 at 16:43