Parabolic subalgebra
A subalgebra of a finite-dimensional Lie algebra over an algebraically closed field that contains a Borel subalgebra, i.e. a maximal solvable subalgebra of
(cf. also Lie algebra, solvable). If
is a finite-dimensional Lie algebra over an arbitrary field
, then a subalgebra
of it is also called a parabolic subalgebra if
is a parabolic subalgebra of
, where
is the algebraic closure of the field
. If
is an irreducible linear algebraic group over a field of characteristic 0 and
is its Lie algebra, then a subalgebra
is a parabolic subalgebra in
if and only if it coincides with the Lie algebra of some parabolic subgroup of
.
Examples of parabolic subalgebras in the Lie algebra of all square matrices of order over a field
are the subalgebras of type
(
is an arbitrary set of natural numbers with sum equal to
), where the algebra
consists of all upper-triangular block-diagonal matrices with as diagonal blocks square matrices of orders
.
Let be a reductive finite-dimensional Lie algebra (cf. Lie algebra, reductive) over a field
of characteristic 0, let
be a maximal diagonalizable subalgebra of
over
, let
be the system of
-roots of
relative to
(cf. Root system), let
be a basis (a set of simple roots) of
, and let
be the group of elementary automorphisms of
, i.e. the group generated by the automorphisms of the form
, where
is a nilpotent element of
. Then every parabolic subalgebra of the Lie algebra
is transformed by some automorphism from
into one of the standard parabolic subalgebras of the type
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where is the centralizer of the subalgebra
in
,
is the root subspace of the Lie algebra
corresponding to the root
,
is an arbitrary subset of the set
, and
is the set of those roots in
whose decomposition into the sum of simple roots from
contains elements of
only with non-negative coefficients. In this way the number of classes of parabolic subalgebras conjugate with respect to
turns out to be
, where
is the
-rank of the semi-simple Lie algebra
. In addition, if
, then
. In particular,
, and
is the minimal parabolic subalgebra of
.
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 |
[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 |
[5] | A. Borel, J. Tits, "Eléments unipotents et sous-groupes paraboliques de groupes réductifs I" Invent. Math. , 12 (1971) pp. 95–104 |
Parabolic subalgebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_subalgebra&oldid=21904