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Let $\mathbb{K}$ be an arbitrary field, and $G$ an algebraic group (=group variety?) over this field. A Borel subgroup of $G$ is a connected solvable subgroup variety $B$ of $G$ such that $G/B$ is complete. A parabolic subgroup of $G$ is a subgroup variety $P$ such that $G/P$ is complete.

Given a group $G$ with lie algebra $\mathfrak{g}$, I would expect that a parabolic Lie subalgebra $\mathfrak{p} \leq \mathfrak{g}$ should be a Lie subalgebra which is the Lie algebra of a parabolic subgroup, and likewise for Borel subalgebras $\mathfrak{b} \leq \mathfrak{g}$.

Since distinct groups can obviously have isomorphic Lie algebras, I would expect that there should be a characterization of Borel and parabolic subalgebras in terms of Lie algebra properties alone, without reference to groups.

However, it seems that in every source I've found that discusses Borel and parabolic subalgebras, that they are defined only for finite-dimensional semisimple and complex lie algebras $\mathfrak{g}$. A Borel subalgebra of a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$ is usually defined to be a maximal solvable subalgebra $\mathfrak{b} \leq \mathfrak{g}$, and a parabolic subalgebra $\mathfrak{p} \leq \mathfrak{g}$ is any subalgebra containing a Borel subalgebra.

Is there a reason for restricting to complex semisimple Lie algebras? Does the definition not work for general Lie algebras?

More generally, what is the relationship between parabolic subgroups and subalgebras? Does the relationship depend at all on the field $\mathbb{K}$ we are working with (characteristic, whether or not it if it is algebraically closed etc.)? Does it depend on the properties of the group (to connected, simply connected, etc.)?

Unfortunately my algebraic geometry knowledge is very minimal. I'm working on stuff in parabolic differential geometry, so I'm hoping to get a better understanding of parabolic subgroups and algebras, especially in the real case.

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    $\begingroup$ One reason for not usually considering these for non-semisimple Lie algebras is that they correspond to ones for the semisimple quotient anyway (as any Borel will contain the radical). $\endgroup$ Aug 2 '16 at 7:52
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    $\begingroup$ Crossposted at MO. $\endgroup$ Aug 3 '16 at 20:42
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There are certainly sources defining Borel subalgebras and parabolic subalgebras in general, e.g., in the book "Lie Algebras and Algebraic Groups" by Patrice Tavel and Rupert W. T. Yu. "We generalise the definitions of Borel subalgebras and parabolic subalgebras to arbitrary Lie algebras and establish the relations between the group objects and the Lie algebra objects".

This contains the details we want (Chapter $29$).

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  • $\begingroup$ Thanks! This is exactly the sort of text I was looking for. Just one question: the authors state that they restrict to algebraically closed fields of characteristic zero. Do you happen to know how their results in chapter 29 concerning the link between parabolic subgroups and subalgebras changes in the non-algebraically closed case? Or for fields of different characteristic? $\endgroup$
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    Aug 2 '16 at 11:52
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    $\begingroup$ Some results are no longer true in characteristic $p$ here, e.g., the results concerning Levi subgroups in connection with parabolic subgroups are problematic. $\endgroup$ Aug 2 '16 at 14:49
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At least for $\mathfrak{g}$ real semisimple there is a nice definition without using maximal solvable Lie subalgebras. A subalgebra $\mathfrak{p}\leq\mathfrak{g}$ is parabolic if $\mathfrak{p}^\perp\leq\mathfrak{p}$ is nilpotent, where $\mathfrak{p}^\perp$ is the polar with respect to the Killing form. I think this definition could be extended to other cases as well.

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Bourbaki, Lie Groups and Algebras, ch. VIII §3 nos. 3--5, uses the following definition. The setting is: $\mathfrak g$ is a reductive Lie algebra over a field $k$ of characteristic $\mathbf{0}$.

A subalgebra $\mathfrak a \subseteq g$ is called Borel / parabolic if, for an algebraic closure $\bar k \vert k$, the subalgebra $\mathfrak a \otimes_k \bar k \subseteq \mathfrak g \otimes_k \bar k$ is Borel / parabolic.

(Where it is used that over an algebraically closed field $\bar k$, Borels and parabolics are defined in one of the well-known ways, equivalently w.r.t. some choice of Cartan subalgebra and appropriate subsets of the corresponding root system.)

So basically a Borel or parabolic is something which after scalar extension to an algebraic closure becomes a Borel or parabolic. A nice side effect of this is that for any scalar extension $K \vert k$, one gets that $\mathfrak a \subseteq \mathfrak g$ is Borel / parabolic if and only if the scalar extension $\mathfrak a_K$ is Borel / parabolic in the scalar extension $\mathfrak g_K$.

Of course what happens now is that many $\mathfrak g$ contain no Borel subalgebras at all ($\mathfrak g$ itself is always a parabolic, but in some cases it's the only parabolic). But that actually becomes part of classifications: Semisimple Lie algebras without proper parabolics are anisotropic (over the reals, "compact"); semisimple Lie algebras which contain some Borel are quasi-split.

While excluding the notoriously troublesome case of positive characteristic, the generality of this definition notably applies to semisimple Lie algebras over the reals, $p$-adics and numbers fields, and should be compared to other definitions for real semisimple ones in other answers. I used it freely in my thesis about semisimple Lie algebras over characteristic $0$ fields, focussing more on the $p$-adic case.

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