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The quotient $G/H$ of a group $G$ by its subgroup $H$ has a $G$-action - every transitive $G$-set is of this form.

However, the quotient space $\mathfrak g/\mathfrak h$ of a Lie algebra $\mathfrak g$ by its subalgebra $\mathfrak h$ is just a vector space. By analogy with the group case, I am trying to figure out whether it is still "special" in some way.

If $\mathfrak g$ is the Lie algebra of a Lie group or an algebraic group $G$ with some representation $V$, and $\mathfrak h$ is the Lie algebra of the stabilizer $H=G^v$ of some vector $v\in V$, then there is a way to identify the tangent space of the orbit $Gv$ at $v$ with the space $\mathfrak gv=\{gv\mid g\in\mathfrak g\}$, so that the surjective map $\mathfrak g\twoheadrightarrow\mathfrak gv$ sending $g$ to $gv$ has kernel $\mathfrak h$. Note that although $\mathfrak g$ acts on $V$, the subspace $\mathfrak gv$ is not in general closed under the $\mathfrak g$-action (and does not in general contain $v$).

So one possibility to relate spaces $\mathfrak g/\mathfrak h$ to quotients like $G/H$ would be to ask whether there exists a $\mathfrak g$-representation $V$ and a vector $v\in V$ such that $\mathfrak h=\{g\in\mathfrak g\mid gv=0\}$ and there is an isomorphism $\mathfrak g/\mathfrak h\cong\mathfrak gv$ compatible with the quotient maps $\mathfrak g\twoheadrightarrow\mathfrak g/\mathfrak h$ sending $g$ to $g+\mathfrak h$ and $\mathfrak g\twoheadrightarrow\mathfrak gv$ sending $g$ to $gv$.

Can this (or maybe something better) be always done?

hm2020
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მამუკა ჯიბლაძე
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    This maybe goes in a different direction than you want, but it's quite natural to use that $\mathfrak g/ \mathfrak h$ is naturally a representation of $\mathfrak h$. I used that recently in my answer here https://math.stackexchange.com/q/3765646/96384, and I remember using it several times in my thesis for classification results (choose $\mathfrak h$ with a well understood representation theory; in a way, the entire Jacobson-Morozov $\mathfrak{sl}_2$-triple business is a development of that theory for the base case of $\mathfrak h = \mathfrak{sl}_2$). – Torsten Schoeneberg Jul 30 '20 at 19:12
  • Not to point out something dumb, but $\mathfrak{g}/\mathfrak{h}$ is the Lie algebra of $G/H$. So, at least in the case when $\mathfrak{g}$ and $\mathfrak{h}$ explicitly come from group theory, there is a concrete interpretation. This shows up, for example, in the theory of symmetric spaces quite apparently. – Alex Youcis Aug 12 '20 at 15:01
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    @AlexYoucis What you say applies when $H$ is a normal subgroup. When not, $G/H$ is not a group and $\mathfrak g/\mathfrak h$ is not a Lie algebra, although it is presumably still true that $\mathfrak g/\mathfrak h$ can be identified with the tangent space of $G/H$ at the origin. – მამუკა ჯიბლაძე Aug 12 '20 at 15:33
  • @მამუკაჯიბლაძე Yes, I meant tangent space at the distinguished point of $G/H$. – Alex Youcis Aug 12 '20 at 16:05

1 Answers1

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I believe the following holds: If $k$ is a field and $H \subseteq G$ is a closed subgroup of a linear algebraic group $G$, it follows the quotient $G/H$ is a smooth quasi projective scheme of finite type over $k$. We may embed $G/H \subseteq \mathbb{P}^d_k$ into projective $d$-space over $k$ for some integer $d \geq 1$ (it is not completely trivial to construct such an embedding). As noted above: The tangent space $T_{\overline{e}}(G/H)$ of the scheme $G/H$ at the point $\overline{e}\in G/H$, where $e\in G$ is the multiplicative identity equals the quotient $Lie(G)/Lie(H)$ as left $H$ and $Lie(H)$-module. There is an isomorphism of $H$-modules (and $Lie(H)$-modules)

$$ T_{\overline{e}}(G/H) \cong Lie(G)/Lie(H).$$

Question: "So one possibility to relate spaces $\mathfrak{g}/\mathfrak{h}$ to quotients like $G/H$ would be to ask whether there exists a $\mathfrak{g}$-representation $V$ and a vector $v \in V$ such that $\mathfrak{h}=\{g \in \mathfrak{g} : gv=0\}$ and there is an isomorphism $\mathfrak{g}/\mathfrak{h}≅\mathfrak{g}v$ compatible with the quotient maps $\mathfrak{g}\rightarrow \mathfrak{g}/\mathfrak{h}$ sending $g$ to $g+h$ and $\mathfrak{g}\rightarrow \mathfrak{g}v$ sending $g$ to $gv$. Can this (or maybe something better) be always done?"

Answer: "This" can be done and you may find it in the litterature. You should take a look at Borel's book "Linear algebraic groups" and Theorem 6.8, page 98. I believe this gives a construction of a quotient $G/H$ for any field $k$. Much work has been done on the construction of "quotients" in algebraic geometry, but Borel's book is a good place to start if your group is algebraic.

You are propably aware of the "stacks homepage" where more general "quotients" such as "algebraic spaces" and "algebraic stacks" are constructed.

Example 1. If $G$ is a semi simple algebraic group and $P\subseteq G$ a parabolic subgroup, it follows the quotient $\mathfrak{g}/\mathfrak{p}$ is the tangent space of the flag variety $G/P$ "at the identity". There is a canonical $\mathfrak{p}$-module structure

$$\rho: \mathfrak{p}\rightarrow \operatorname{End}(\mathfrak{g}/\mathfrak{p})$$

induced by the canonical action - the adjoint action - of $\mathfrak{p}$ on $\mathfrak{g}$.

Example 2. If $dim_k(V)=n+1$ and $l\in V$ is a line, you may let $P\subseteq SL(V)$ be the parabolic subgroup of elements fixing the line $l$. It follows there is an isomorphism

$$SL(V)/P \cong \mathbb{P}(V^*)\cong \mathbb{P}^n_k.$$

Hence the isomorphism

$$ T_{\overline{e}}(SL(V)/P) \cong T_{\overline{e}}(\mathbb{P}^n_k)$$

gives a canonical $\mathfrak{p}$-module structure on the tangent space of projective $n$-space at the "identity". If you instead choose a vector subspace $W\subseteq V$ of dimension $m$ you get a similar result for the grassmannian. You get a geometric interpretation of these representations in terms of flag varieties and other quotient spaces.

Example 3. If $V:=k\{e_0,..,e_n\}$, $l:=(e_0)$, $V^*:=k\{x_0,..,x_n\}$ and $\mathcal{L}:=\mathcal{O}(d)$ for $d\geq 1$, it follows there is an isomorphism

$$ \phi:\Gamma(\mathbb{P}^n_k, \mathcal{L}) \cong Sym^d(V^*)$$

where $Sym^d(V^*)$ is the $d$'th symmetric product of the dual representation $V^*$ of $V$. The map $\phi$ is a map of $SL(V)$-modules. Hence the global sections of $\mathcal{L}$ is an (irreducible) $SL(V)$-module. Hence you may interpret some irreducible $SL(V)$-modules as global sections of invertible sheaves on the quotient $SL(V)/P$. This is the Borel-Weil-Bott construction.

hm2020
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