Borel subalgebras

E542124

mathematical concept notion in Lie theory notion in representation theory

Borel subalgebras are maximal solvable subalgebras of a Lie algebra that play a central role in the classification and representation theory of Lie algebras and algebraic groups.

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Observed surface forms (1)

Surface form	Occurrences
Borel subgroups	1

Statements (47)

Predicate	Object
instanceOf	mathematical concept ⓘ notion in Lie theory ⓘ notion in representation theory ⓘ
assumption	often considered over algebraically closed fields of characteristic zero ⓘ
classificationRole	choice of Borel and Cartan subalgebras determines simple roots ⓘ choice of Borel subalgebra determines a system of positive roots ⓘ used to define the Weyl chamber decomposition ⓘ
component	have nilradical equal to the sum of positive root spaces in the semisimple case ⓘ
conjugacy	any two Borel subalgebras of a complex semisimple Lie algebra are conjugate under the adjoint group ⓘ
construction	can be constructed as the direct sum of a Cartan subalgebra and positive root spaces ⓘ
containment	contain a Cartan subalgebra ⓘ contain all positive root spaces relative to a choice of positive roots ⓘ
context	finite-dimensional Lie algebras over an algebraically closed field of characteristic zero ⓘ
definition	maximal solvable subalgebras of a Lie algebra ⓘ
example	the subalgebra of upper triangular matrices in gl_n(C) is a Borel subalgebra ⓘ
exampleOf	solvable subalgebras that are not nilpotent in general ⓘ
existence	every finite-dimensional complex semisimple Lie algebra has Borel subalgebras ⓘ
field	Lie algebras ⓘ algebraic groups ⓘ
generalization	have analogues in Kac–Moody algebras ⓘ have analogues in real semisimple Lie algebras via minimal parabolic subalgebras ⓘ
geometricInterpretation	are stabilizers of complete flags in the standard representation for classical groups ⓘ correspond to points of the flag variety ⓘ
historicalNote	introduced and systematically studied by Armand Borel in the context of algebraic groups ⓘ
maximality	are maximal among solvable subalgebras but not necessarily maximal among all subalgebras ⓘ
namedAfter	Armand Borel NERFINISHED ⓘ
property	are self-normalizing in a semisimple Lie algebra ⓘ maximal with respect to inclusion among solvable subalgebras ⓘ solvable ⓘ their normalizer equals themselves in a semisimple Lie algebra ⓘ
relation	are Lie algebras of Borel subgroups ⓘ are minimal parabolic subalgebras in the complex semisimple case ⓘ contain maximal nilpotent subalgebras (nilradicals) ⓘ correspond to Borel subgroups in algebraic groups ⓘ intersections of distinct Borel subalgebras often equal a Cartan subalgebra in the semisimple case ⓘ
role	central in the classification of complex semisimple Lie algebras ⓘ central in the representation theory of semisimple Lie algebras ⓘ central in the structure theory of semisimple Lie algebras ⓘ central in the theory of algebraic groups ⓘ used in the classification of irreducible finite-dimensional representations ⓘ used to define Verma modules ⓘ used to define highest weight representations ⓘ used to define the notion of highest weight of a representation ⓘ
use	used in the Borel–Weil and Borel–Weil–Bott theorems ⓘ used in the proof of the classification of root systems ⓘ used to define parabolic subalgebras as subalgebras containing a Borel subalgebra ⓘ used to define the flag variety as the set of all Borel subalgebras containing a fixed Cartan subalgebra ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lie theory → studies → Borel subalgebras ⓘ

Cartan subalgebras → relatedTo → Borel subalgebras ⓘ

Deligne–Lusztig theory → involves → Borel subalgebras ⓘ

this entity surface form: Borel subgroups