View source for Parabolic subalgebra

A subalgebra of a finite-dimensional [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712601.png" /> over an algebraically closed field that contains a Borel subalgebra, i.e. a maximal solvable subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712602.png" /> (cf. also [[Lie algebra, solvable|Lie algebra, solvable]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712603.png" /> is a finite-dimensional Lie algebra over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712604.png" />, then a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712605.png" /> of it is also called a parabolic subalgebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712606.png" /> is a parabolic subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712608.png" /> is the algebraic closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p0712609.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126010.png" /> is an irreducible [[Linear algebraic group|linear algebraic group]] over a field of characteristic 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126011.png" /> is its Lie algebra, then a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126012.png" /> is a parabolic subalgebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126013.png" /> if and only if it coincides with the Lie algebra of some [[Parabolic subgroup|parabolic subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126014.png" />.

Examples of parabolic subalgebras in the Lie algebra of all square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126015.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126016.png" /> are the subalgebras of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126017.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126018.png" /> is an arbitrary set of natural numbers with sum equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126019.png" />), where the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126020.png" /> consists of all upper-triangular block-diagonal matrices with as diagonal blocks square matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126021.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126022.png" /> be a reductive finite-dimensional Lie algebra (cf. [[Lie algebra, reductive|Lie algebra, reductive]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126023.png" /> of characteristic 0, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126024.png" /> be a maximal diagonalizable subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126025.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126027.png" /> be the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126028.png" />-roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126029.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126030.png" /> (cf. [[Root system|Root system]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126031.png" /> be a basis (a set of simple roots) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126032.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126033.png" /> be the group of elementary automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126034.png" />, i.e. the group generated by the automorphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126036.png" /> is a nilpotent element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126037.png" />. Then every parabolic subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126038.png" /> is transformed by some automorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126039.png" /> into one of the standard parabolic subalgebras of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126040.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126041.png" /> is the [[Centralizer|centralizer]] of the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126044.png" /> is the root subspace of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126045.png" /> corresponding to the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126047.png" /> is an arbitrary subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126049.png" /> is the set of those roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126050.png" /> whose decomposition into the sum of simple roots from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126051.png" /> contains elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126052.png" /> only with non-negative coefficients. In this way the number of classes of parabolic subalgebras conjugate with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126053.png" /> turns out to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126055.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126056.png" />-rank of the semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126057.png" />. In addition, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126059.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126061.png" /> is the minimal parabolic subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071260/p07126062.png" />.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII {{MR|0682756}} {{MR|0573068}} {{MR|0271276}} {{MR|0240238}} {{MR|0132805}} {{ZBL|0329.17002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Morozov, "Proof of the regularity theorem" ''Uspekhi Mat. Nauk'' , '''11''' (1956) pp. 191–194 (In Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G.D. Mostow, "On maximal subgroups of real Lie groups" ''Ann. of Math.'' , '''74''' (1961) pp. 503–517 {{MR|0142687}} {{ZBL|0109.02301}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Borel, J. Tits, "Eléments unipotents et sous-groupes paraboliques de groupes réductifs I" ''Invent. Math.'' , '''12''' (1971) pp. 95–104 {{MR|0294349}} {{ZBL|0238.20055}} </TD></TR></table>