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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). $\endgroup$ – Tobias Kildetoft Aug 2 '16 at 7:52
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    $\begingroup$ Crossposted at MO. $\endgroup$ – Dietrich Burde 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? $\endgroup$ – ಠ_ಠ 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. $\endgroup$ – Dietrich Burde 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 $\mathfrak{p}^\perp$ is nilpotent, where $\mathfrak{p}^\perp$ is the polar with respect to the Killing form. Then also $\mathfrak{p}^\perp\leq\mathfrak{p}$. I think this definition could be extended to other cases as well.

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