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Type of subgroup of an algebraic group
algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the
Borel_subgroup
Group with subnormal series where all factors are abelian
two of the Borel subgroups. The example given above, the subgroup B {\displaystyle B} in G L 2 {\displaystyle GL_{2}} , is a Borel subgroup. In G L 3 {\displaystyle
Solvable_group
Swiss mathematician (1923–2003)
452) Borel–Weil–Bott theorem Borel cohomology Borel conjecture Borel construction Borel subgroup Borel subalgebra Borel fixed-point theorem Borel's theorem
Armand_Borel
Subgroup of the group of invertible n×n matrices
called the Borel subgroup of G L ( n ) {\displaystyle GL(n)} . It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of G L
Linear_algebraic_group
Special kind of square matrix
algebra. These are, respectively, the standard Borel subgroup B of the Lie group GLn and the standard Borel subalgebra b {\displaystyle {\mathfrak {b}}}
Triangular_matrix
Mathematical group
parabolic subgroup is any subgroup that is a standard parabolic relative to some Borel subgroup, or equivalently (since all Borel subgroups are conjugate)
Parabolic subgroup of a reflection group
Parabolic_subgroup_of_a_reflection_group
Basic result in the representation theory of Lie groups
{\displaystyle \mathbb {C} } , and fix a maximal torus T along with a Borel subgroup B which contains T. Let λ be an integral weight of T; λ defines in a
Borel–Weil–Bott_theorem
Topics referred to by the same term
Parabolic subgroup may refer to: a parabolic subgroup of a reflection group a subgroup of an algebraic group that contains a Borel subgroup This disambiguation
Parabolic_subgroup
Concept in mathematics
it contains a Borel subgroup over k. A split reductive group is quasi-split. If G is quasi-split over k, then any two Borel subgroups of G are conjugate
Reductive_group
Special group in linear algebra
an Iwahori subgroup is a subgroup of a reductive algebraic group over a nonarchimedean local field that is analogous to a Borel subgroup of an algebraic
Iwahori_subgroup
the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Let g = g l ( V ) {\displaystyle {\mathfrak {g}}={\mathfrak
Borel_subalgebra
Topics referred to by the same term
after Émile Borel Borel subgroup, in the theory of algebraic groups, named after Armand Borel Borel (surname), a surname Etablissements Borel, an aircraft
Borel
Linear representation in mathematics
parabolic subgroups containing a Borel subgroup, of the representation induced from the identity representation of the parabolic subgroup. The Steinberg
Steinberg_representation
Maximal connected Abelian subgroup
subgroup is a Cartan subgroup. Borel subgroup Algebraic group Algebraic torus Milne (2017), Proposition 17.44. Milne (2017), Corollary 17.84. Borel,
Cartan_subgroup
Algebraic variety with a group structure
variety Borel subgroup Tame group Morley rank Cherlin–Zilber conjecture Adelic algebraic group Pseudo-reductive group Borel 1991, p.54. Borel 1991, p
Algebraic_group
variety of Borel subgroups B, then the Springer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to
Springer_resolution
Technique in mathematical group theory
parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to G). The representations of parabolic induction
Deligne–Lusztig_theory
Baily and Armand Borel (1964, 1966). If C is the quotient of the upper half plane by a congruence subgroup of SL2(Z), then the Baily–Borel compactification
Baily–Borel_compactification
Concept in group theory
the rank. We call B the (standard) Borel subgroup, T the (standard) Cartan subgroup, and W the Weyl group. A subgroup of G is called parabolic if it contains
(B,_N)_pair
Universal construction of a complex Lie group from a real Lie group
into the Borel subgroup of lower triangular matrices in GC. The Bruhat decomposition is easy to prove for SL(n,C). Let B be the Borel subgroup of upper
Complexification_(Lie_group)
Theorem in algebraic geometry
closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem,
Kempf_vanishing_theorem
Mathematical group
suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may
Outer_automorphism_group
In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is
Spherical_variety
Linear algebraic group
mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field
Quasi-split_group
Infinite family of simple groups of Lie type
least 4 types of maximal subgroups. The diagonal subgroup is cyclic, of order q – 1. The lower triangular (Borel) subgroup and its conjugates, of order
Suzuki_groups
Subgroup of a root system's isometry group
cannot always be realized as a subgroup of G. If B is a Borel subgroup of G, i.e., a maximal connected solvable subgroup and a maximal torus T = T0 is
Weyl_group
Group that is also a differentiable manifold with group operations that are smooth
subgroup of G {\displaystyle G} admits a unique smooth structure which makes it an embedded Lie subgroup of G {\displaystyle G} —i.e. a Lie subgroup such
Lie_group
Algebraic geometry analog of a principal bundle in algebraic topology
P\times _{X_{R}}X_{R'}} admits a reduction of structure group scheme to a Borel subgroup-scheme of G {\displaystyle G} . It is common to consider a torsor for
Torsor_(algebraic_geometry)
Mathematical structure
pairs and calling any conjugate of B a Borel subgroup and any group containing a Borel subgroup a parabolic subgroup, the vertices of the building X correspond
Building_(mathematics)
Algebraic geometry theorem
{\displaystyle X=\mathbb {P} ^{n}} is expressed as the quotient of SLn by the Borel subgroup of upper triangular matrices, Z is a subvariety and Y is a hyperplane
Theorem_of_Bertini
Concept in mathematics
group, and λ:H→C a character on a maximal torus H. It extends to a Borel subgroup λ:B→C, giving a one dimensional representation Wλ of B. Then GxWλ is
Equivariant_sheaf
Classification theorem in group theory
configuration occurs in the group SL2(2q), with PU a Borel subgroup of upper triangular matrices and Q the subgroup of order 3 generated by y = ( 0 1 1 1 ) {\displaystyle
Feit–Thompson_theorem
Chevalley group over a finite field with pk elements, and B is its Borel subgroup. Iwahori showed that the Hecke ring H(G//B) is obtained from the generic
Hecke_algebra_of_a_pair
field k {\displaystyle k} is quasi-split if and only if it admits a Borel subgroup B ⊆ G {\displaystyle B\subseteq G} defined over k {\displaystyle k}
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Rational function of the form (az + b)/(cz + d)
\mathbb {C} \right\};} this is an example of the unipotent radical of a Borel subgroup (of the Möbius group, or of SL(2, C) for the matrix group; the notion
Möbius_transformation
algebraic group G {\displaystyle G} with a Borel subgroup B {\displaystyle B} and a standard parabolic subgroup P {\displaystyle P} , it is known that the
Schubert_variety
Deformation of the group algebra of a Coxeter group
Chevalley group over a finite field with pk elements, and B is its Borel subgroup. Iwahori showed that the Hecke ring H(G//B) is obtained from the generic
Iwahori–Hecke_algebra
(2009). Let G be a reductive group over the complex numbers, and B a Borel subgroup. Then there is an equivalence of categories D -Mod ( G / B ) ≃ (
Beilinson–Bernstein localization
Beilinson–Bernstein_localization
Sequence of spaces in linear algebra
stabilizer subgroup of any complete flag is a Borel subgroup (of the general linear group), and the stabilizer of any partial flags is a parabolic subgroup. The
Flag_(linear_algebra)
Unitary representations of a Lie group
SL(2, R), there is up to conjugacy only one proper parabolic subgroup, the Borel subgroup of the upper-triangular matrices of determinant 1. The inducing
Representation theory of SL2(R)
Representation_theory_of_SL2(R)
Poisson manifold that is also a Lie group
G {\displaystyle B_{\pm }\subset G} be the corresponding opposite Borel subgroups, so that T = B − ∩ B + {\displaystyle T=B_{-}\cap B_{+}} and there
Poisson–Lie_group
Theorem in the representation theory of linear algebraic groups
in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(n,K): the image is simultaneously
Lie–Kolchin_theorem
Principal connection
complex plane C {\displaystyle \mathbb {C} } , with a distinguished Borel subgroup B = B G ⊂ G {\displaystyle B=B_{G}\subset G} . Set N = [ B , B ] {\displaystyle
Oper_(mathematics)
Mathematical term
group over an algebraically closed field. B {\displaystyle B} is a Borel subgroup of G {\displaystyle G} W {\displaystyle W} is a Weyl group of G {\displaystyle
Bruhat_decomposition
Type of mathematical space
group; the set of lower triangular matrices of determinant one is a Borel subgroup. If the field F is the real or complex numbers we can introduce an inner
Generalized_flag_variety
Hungarian mathematician
Pál Pálfy, was about group theory and was entitled "On the Sylow and Borel subgroups of classical groups". After temporary positions at the Alfréd Rényi
Balázs_Szegedy
In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal
Borel–de_Siebenthal_theory
Additive group Multiplicative group Algebraic torus Reductive group Borel subgroup Radical of an algebraic group Unipotent radical Lie–Kolchin theorem
List of algebraic geometry topics
List_of_algebraic_geometry_topics
(1974). Let G be a connected reductive complex algebraic group, B a Borel subgroup and T a maximal torus contained in B. Let w ∈ W = N G ( T ) / T . {\displaystyle
Bott–Samelson_resolution
Class of commutative rings
{\displaystyle G} a reductive group such as G L n {\displaystyle GL_{n}} with Borel subgroups B ± {\displaystyle B_{\pm }} then on G u , v = B u B ∩ B − v B − {\displaystyle
Cluster_algebra
Each semi-simple algebraic group is geometrically reductive
line bundle over G/B corresponding to a character μ of T, where B is a Borel subgroup containing T. If n is sufficiently large then Enρ has dimension (n+1)N
Haboush's_theorem
Mathematical group
matrices have determinant 1 {\displaystyle 1} , the symplectic group is a subgroup of the special linear group SL ( 2 n , F ) {\displaystyle \operatorname
Symplectic_group
Pro-finite group Classification of finite simple groups Alternating group Borel subgroup Chevalley group Conway group Feit–Thompson theorem Fischer group General
List_of_group_theory_topics
In mathematics, representation of a reductive algebraic group
If G is a split reductive group and U is the unipotent radical of a Borel subgroup B, then a Whittaker model for a representation is an embedding of it
Whittaker_model
{\displaystyle G=BWB} of a semisimple algebraic group into double cosets of a Borel subgroup can be regarded as a generalization of the principle of Gauss–Jordan
Lie_group_decomposition
Integral polynomial
of complex flag manifolds G/B where G is a complex Lie group and B a Borel subgroup. The original (K-L) case is then about the details of decomposing B
Kazhdan–Lusztig_polynomial
{n}}^{+}} to a closed subgroup U ⊂ G {\displaystyle U\subset G} . The Lie subgroup B ⊂ G {\displaystyle B\subset G} corresponding to the Borel subalgebra b =
Complex_Lie_algebra
Manifold with inversion symmetry
the classification can be deduced from Borel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric
Hermitian_symmetric_space
Concept in topology
proof of the existence and uniqueness of a maximal compact subgroup can be found in Borel (1950) and Helgason (1978). Cartier (1955) and Hochschild (1965)
Maximal_compact_subgroup
Type of group in group theory
defined as a discrete subgroup with finite covolume. The terminology introduced above is coherent with this, as a theorem due to Borel and Harish-Chandra
Arithmetic_group
isomorphic to the spaces of sections F(λ) of line bundles λ over G/B for a Borel subgroup B. In characteristic 0 this is automatically true as the irreducible
Good_filtration
Group of unitary complex matrices with determinant of 1
operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group U(n), consisting of all n × n unitary matrices. As
Special_unitary_group
Mathematical object
holds: There exists an open (B, K)-double coset in G, where B is the Borel subgroup of G. There is a finite number of (B, K)-double coset in G. For any
Gelfand_pair
2003), a Swiss mathematician 2. A Borel subgroup. 3. A Borel subalgebra is a maximal solvable subalgebra. 4. Borel-Bott-Weil theorem Bruhat 1. Bruhat
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Discrete subgroup in a locally compact topological group
it has a subgroup Γ = G ∩ G L n ( Z ) {\displaystyle \Gamma =G\cap \mathrm {GL} _{n}(\mathbb {Z} )} . A fundamental theorem of Armand Borel and Harish-Chandra
Lattice_(discrete_subgroup)
subgroup Kleinian group Discrete Heisenberg group Clifford–Klein form Borel subgroup Arithmetic group Dunkl operator Modular form Langlands program Sophus
List_of_Lie_groups_topics
Group of 𝑛 × 𝑛 invertible matrices
F)} or SL n ( F ) {\displaystyle \operatorname {SL} _{n}(F)} , is the subgroup of GL ( n , F ) {\displaystyle \operatorname {GL} (n,F)} consisting of
General_linear_group
where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borel showed that its cohomology ring is isomorphic to the quotient
Kostant_polynomial
Topics referred to by the same term
when G is the finite Chevalley group over a finite field and K is its Borel subgroup. Affine Hecke algebra This disambiguation page lists mathematics articles
Hecke algebra (disambiguation)
Hecke_algebra_(disambiguation)
Locally compact topological group with an invariant averaging operation
paradox construction is to find inside the rotation group SO(3) a free subgroup on two generators. Amenable groups cannot contain such groups, and do not
Amenable_group
Type of mathematical space
Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904). The Heine–Borel theorem
Compact_space
Representation theory
irreducible spherical principal series of G induced from the character of the Borel subgroup of G corresponding to λ; these representations are irreducible and can
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Group of matrices with determinant 1
ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det
Special_linear_group
subgroup Gx is a Borel measurable unitary representation U of Gx on H (Here U(H) has the strong operator topology). However, it is known that a Borel
System_of_imprimitivity
Group of unitary matrices
group is a subgroup of the general linear group GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} , and it has as a subgroup the special
Unitary_group
248-dimensional exceptional simple Lie group
group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism group
E8_(mathematics)
Compositio Mathematica, 3: 408–426 Borel, Armand; Wallach, Nolan (2000), Continuous cohomology, discrete subgroups, and representations of reductive groups
Totally_disconnected_group
top-dimensional l-adic cohomology groups of the algebraic variety Bu of the Borel subgroups of G containing a given unipotent element u of a semisimple algebraic
Springer_correspondence
Isometry group of Euclidean space
preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(n) and E+(n), whose elements
Euclidean_group
Set of quantum operations
+ O ( n ) {\displaystyle 2^{n\log(n)+O(n)}} elements. Borel group, a maximal solvable subgroup, which is generated by the product of the lower triangular
Clifford_group
Left-invariant (or right-invariant) measure on locally compact topological group
subsets of G {\displaystyle G} is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g {\displaystyle g} is an element of
Haar_measure
Group of flat spacetime symmetries
subgroup, while the six-dimensional Lorentz group is also a subgroup, the stabilizer of the origin. The Poincaré group itself is the minimal subgroup
Poincaré_group
Concept in topology
two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable
Polish_space
a standard Borel space X is a Borel equivalence relation E with countable classes, that can, in a certain sense, be approximated by Borel equivalence
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
Group theory theorem
closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of
Closed-subgroup_theorem
Group that is a topological space with continuous group operations
H, which are open. If H is a subgroup of G, then the closure of H is also a subgroup. Likewise, if H is a normal subgroup of G, the closure of H is normal
Topological_group
Topics referred to by the same term
points Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure Lattice (module)
Lattice
Descriptive set theory relation
invariant descriptive set theory, countable Borel relations are a class of relations between standard Borel space which are particularly well behaved.
Countable_Borel_relation
Mathematical concept
arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not
Shimura_variety
defined over k. The centralizer of a maximal torus is called a Cartan subgroup. Diagonal subgroup Borel, A. Linear algebraic groups, 2nd ed. v t e v t e
Diagonalizable_group
Set with associative invertible operation
Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year
Group_(mathematics)
Simple Lie group; the automorphism group of the octonions
the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional
G2_(mathematics)
Nilpotent subalgebra of a Lie algebra
Lie algebras and their Representations Infinite-dimensional Lie algebras Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics,
Cartan_subalgebra
Lie group of complex numbers of unit modulus; topologically a circle
group is equal to 1. This integral is not arbitrary, but is the unique (Borel) probability measure that is invariant under constant change of variables
Circle_group
Russian mathematician
It had been known since the 1950s (Borel, Harish-Chandra) that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces
Grigory_Margulis
Concept in class field theory
(where the superscript c {\displaystyle c} denotes the commutator subgroup). The Weil group of a class formation with fundamental classes u E / F
Weil_group
American mathematician
representations are of type I) if and only if the Borel structure of its dual is a standard Borel space. He has written numerous survey articles connecting
George_Mackey
discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of
Ergodic_flow
{\displaystyle {\mathfrak {p}}} contains a maximal solvable subalgebra (a Borel subalgebra) of g {\displaystyle {\mathfrak {g}}} ; the orthogonal complement
Parabolic_Lie_algebra
Lie group of Lorentz transformations
negligible, physical laws are Lorentz-invariant. The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime
Lorentz_group
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