# Parabolic subalgebra

A subalgebra of a finite-dimensional Lie algebra $\mathfrak g$ over an algebraically closed field that contains a Borel subalgebra, i.e. a maximal solvable subalgebra of $\mathfrak g$ ( cf. also Lie algebra, solvable). If $\mathfrak g$ is a finite-dimensional Lie algebra over an arbitrary field $k$ , then a subalgebra $\mathfrak p$ of it is also called a parabolic subalgebra if $\mathfrak p \otimes _{k} \overline{k}$ is a parabolic subalgebra of $\mathfrak g \otimes _{k} \overline{k}$ , where $\overline{k}$ is the algebraic closure of the field $k$ . If $G$ is an irreducible linear algebraic group over a field of characteristic 0 and $\mathfrak g$ is its Lie algebra, then a subalgebra $\mathfrak p \subset \mathfrak g$ is a parabolic subalgebra in $\mathfrak g$ if and only if it coincides with the Lie algebra of some parabolic subgroup of $G$ .

Examples of parabolic subalgebras in the Lie algebra of all square matrices of order $n$ over a field $k$ are the subalgebras of type $\mathfrak p ( \mu )$ ( $\mu = (m _{1} \dots m _{s} )$ is an arbitrary set of natural numbers with sum equal to $n$ ), where the algebra $\mathfrak p ( \mu )$ consists of all upper-triangular block-diagonal matrices with as diagonal blocks square matrices of orders $m _{1} \dots m _{s}$ .

Let $\mathfrak g$ be a reductive finite-dimensional Lie algebra (cf. Lie algebra, reductive) over a field $k$ of characteristic 0, let $\mathfrak f$ be a maximal diagonalizable subalgebra of $\mathfrak g$ over $k$ , let $R$ be the system of $k$ - roots of $\mathfrak g$ relative to $\mathfrak f$ ( cf. Root system), let $\Delta$ be a basis (a set of simple roots) of $R$ , and let $\mathop{\rm Aut}\nolimits _{e} \ \mathfrak g$ be the group of elementary automorphisms of $\mathfrak g$ , i.e. the group generated by the automorphisms of the form $\mathop{\rm exp}\nolimits \ \mathop{\rm ad}\nolimits \ x$ , where $x$ is a nilpotent element of $\mathfrak g$ . Then every parabolic subalgebra of the Lie algebra $\mathfrak g$ is transformed by some automorphism from $\mathop{\rm Aut}\nolimits _{e} \ \mathfrak g$ into one of the standard parabolic subalgebras of the type $$\mathfrak p _ \Phi = \mathfrak g ^{0} + \sum _ {\alpha \in \Pi ( \Phi )} \mathfrak g ^ \alpha ,$$ where $\mathfrak g ^{0}$ is the centralizer of the subalgebra $\mathfrak f$ in $\mathfrak g$ , $\mathfrak g ^ \alpha$ is the root subspace of the Lie algebra $\mathfrak g$ corresponding to the root $\alpha \in R$ , $\Phi$ is an arbitrary subset of the set $\Delta$ , and $\Pi ( \Phi )$ is the set of those roots in $R$ whose decomposition into the sum of simple roots from $\Delta$ contains elements of $\Phi$ only with non-negative coefficients. In this way the number of classes of parabolic subalgebras conjugate with respect to $\mathop{\rm Aut}\nolimits _{e} \ \mathfrak g$ turns out to be $2 ^{r}$ , where $r = | \Delta |$ is the $k$ - rank of the semi-simple Lie algebra $[ \mathfrak g ,\ \mathfrak g]$ . In addition, if $\Phi _{1} \subseteq \Phi _{2}$ , then $\mathfrak p _ {\Phi _{1}} \supseteq \mathfrak p _ {\Phi _{2}}$ . In particular, $\mathfrak p _ \emptyset = \mathfrak g$ , and $\mathfrak p _ \Delta$ is the minimal parabolic subalgebra of $\mathfrak g$ .

All non-reductive maximal subalgebras of finite-dimensional reductive Lie algebras over a field of characteristic 0 are parabolic subalgebras (see [2], [3], [5]).

#### References

 [1] N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII MR0682756 MR0573068 MR0271276 MR0240238 MR0132805 Zbl 0329.17002 [2] F.I. Karpelevich, "On non-semi-simple maximal subalgebras of semi-simple Lie algebras" Dokl. Akad. Nauk USSR , 76 (1951) pp. 775–778 (In Russian) [3] V.V. Morozov, "Proof of the regularity theorem" Uspekhi Mat. Nauk , 11 (1956) pp. 191–194 (In Russian) [4] G.D. Mostow, "On maximal subgroups of real Lie groups" Ann. of Math. , 74 (1961) pp. 503–517 MR0142687 Zbl 0109.02301 [5] A. Borel, J. Tits, "Eléments unipotents et sous-groupes paraboliques de groupes réductifs I" Invent. Math. , 12 (1971) pp. 95–104 MR0294349 Zbl 0238.20055
How to Cite This Entry:
Parabolic subalgebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_subalgebra&oldid=44280
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article