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