In algebra, a parabolic Lie algebra p {\displaystyle {\mathfrak {p}}} is a subalgebra of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} satisfying...

positive weight. A Lie subalgebra of g {\displaystyle {\mathfrak {g}}} containing a Borel subalgebra is called a parabolic Lie algebra. Hyperbolic group...

parabolic Lie algebra is also a graded Lie algebra. A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket...

function Parabolic induction Parabolic Lie algebra Parabolic partial differential equation In physics: Parabolic trajectory In technology: Parabolic antenna...

Monster Lie algebra Nilpotent Lie algebra Nilradical of a Lie algebra Orthogonal symmetric Lie algebra Parabolic Lie algebra Pre-Lie algebra Quadratic Lie algebra...

In linear algebra, the trace of a square matrix A, denoted tr(A), is the sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle...

minimal parabolic subgroup of G, and N = N(K), where N is the normalizer of a split maximal torus contained in P. In particular, any finite group of Lie type...

Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of g, this semisimple Lie...

consisting entirely of parabolics, X 1 , X 3 {\displaystyle X_{1},X_{3}} generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group...

tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated...

algebraic or Lie group, then the (generalized) flag variety for G is G/P where P is a parabolic subgroup of G. The correspondence between parabolic subgroups...

is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic...

geometry. The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding...

of a Lie algebra g {\displaystyle {\mathfrak {g}}} is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra g {\displaystyle...

a reductive algebraic group and P = M A N {\displaystyle P=MAN} is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists...

over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups...

Cartan subgroup, group of adjoint type, parabolic induction Real form (Lie theory), Satake diagram Adelic algebraic group, Weil's conjecture on Tamagawa...

{\displaystyle {\mathfrak {g}}_{1}} is the Levi component of a parabolic subalgebra of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . For a given dominant...

Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances. For example, the Lie...

theory of algebraic groups, through their connection with Weyl groups. The parabolic subgroups are so-named because they correspond to parabolic subgroups...

basis H, X, Y for the complexification of the Lie algebra of SL(2, R) so that iH generates the Lie algebra of a compact Cartan subgroup K (so in particular...

{\displaystyle {\mathfrak {n}}.} This algebra is the derived Lie algebra of b {\displaystyle {\mathfrak {b}}} , the Lie algebra of all upper triangular matrices;...

In the theory of Lie algebras, an sl2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators...

correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group. Let G be connected compact Lie group with...

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing"...

and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra g {\displaystyle...

mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation...

anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important...

theory of parabolic geometries. Let g {\displaystyle {\mathfrak {g}}} be a semisimple Lie algebra and p {\displaystyle {\mathfrak {p}}} a parabolic subalgebra...

for the matrix group; the notion is defined for any reductive Lie group). All non-parabolic transformations have two fixed points and are defined by a matrix...