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I have a question on quotients of linear algebraic groups. Let $G$ be a linear algebraic group and $H$ a linear algebraic group acting on $G$ as an algebraic group.

I would like to know what the quotient $G/H$ is, in several categories (i.e. we have the forgetful functor from linear algebraic groups to (including but not only) $(Sets)$, $(Groups)$, $(affine-schemes)$, etc.)

My question is: in which of those categories does such a quotient exist (under which assumptions) and when does taking the quotient commute with the several forgetful functors? Further, is any of that easy to see?

I know this question is quite open (and some things are also clear, e.g. for the quotient to exist in Groups we should ask $H$ to act as a normal divisor of $G$), but I'd be happy about any insight.

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  • $\begingroup$ I think a more apropos question is what you mean by quotient. There are various definitions even for $H$ a subgroup of $G$: naive quotients (group representing $T\mapsto G(T)/H(T)$), categorical quotient (group with $H$-invariant map, which is initial with respect to this data), and geometric quotient (a la GIT by Mumford). $\endgroup$ – Alex Youcis Oct 30 '14 at 22:23
  • $\begingroup$ I was thinking about the categorical quotient (i.e. $G \rightarrow G/H$ such that a morphism $G \rightarrow X$ factorizes over $G/H$ if and only if it is $H$-invariant). I hope I got the universal property right. $\endgroup$ – Louis Oct 30 '14 at 22:53
  • $\begingroup$ I can talk about a specific but important example. If you take the standard action of $H$ on $G$ via left multiplication, then the quotient always exists in affine schemes. Not only is this a categorical quotient, but it is in fact a geometric quotient, which has the following much nicer properties: the map from $G$ to $G/H$ is flat and $H$ invariant and the natural map $G \times_{k} H \rightarrow G \times_{G/H} G$ is an isomorphism. $\endgroup$ – Siddharth Venkatesh Oct 31 '14 at 7:43
  • $\begingroup$ I think in affine schemes the categorical quotient always exists (namely take the spectrum of the maximal $H$-invariant subring of $\Gamma (G)$). But of particular interest for me is, when is this also a linear algebraic group and when does the group structure on this group agree with the one on the quotient in groups. $\endgroup$ – Louis Oct 31 '14 at 9:54

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