In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can...

known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. Bott was born in Budapest...

In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For...

 452) Borel–Weil–Bott theorem Borel cohomology Borel conjecture Borel construction Borel subgroup Borel subalgebra Borel fixed-point theorem Borel's theorem...

Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results...

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Bott, character in a Richmal Crompton novel. Wilf Bott (1907–1992), English footballer Atiyah–Bott fixed-point theorem Borel–Weil–Bott theorem Bott periodicity...

varieties may be studied using complex geometry leading to the Borel–Weil–Bott theorem, or in symplectic geometry, where Kähler manifolds are symplectic...

integer multiples of ⁠ 2 π {\displaystyle 2\pi } ⁠. By the first isomorphism theorem we then have that T ≅ R   /   2 π Z . {\displaystyle \mathbb {T} \cong...

chamber associated to the indicated base. A basic general theorem about Weyl chambers is this: Theorem: The Weyl group acts freely and transitively on the Weyl...

studied in representation theory. In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning...

corresponding connected Lie group, unique up to covering spaces (Lie's third theorem). This correspondence allows one to study the structure and classification...

adding one dimension at a time. A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical...

of G. This is the physical interpretation of the Borel–Weil theorem or the Borel–Weil–Bott theorem. The Lagrangian of these theories is the classical...

algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem Lie groups in physics Particle physics and representation...

In mathematics, the 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...

written as the quotient space SO+(1, 3) / SO(3), due to the orbit-stabilizer theorem. Furthermore, this upper sheet also provides a model for three-dimensional...

{\displaystyle m} are not faithful. See under the example for Borel–Weil–Bott theorem. Representations of SU(2) describe non-relativistic spin, due to...

(with the usual identifications). It follows from the inverse function theorem that the exponential map, therefore, restricts to a diffeomorphism from...

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, but unlike...

algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem Lie groups in physics Particle physics and representation...

algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem Lie groups in physics Particle physics and representation...

centralizer of the identity component G0 of G. By the first isomorphism theorem we have A d ( G ) ≅ G / Z G ( G 0 ) . {\displaystyle \mathrm {Ad} (G)\cong...

JSTOR 1969129. Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups, American Mathematical Society, ISBN 978-0-8218-0288-5. Borel, Armand;...

spacetime dimensions) associated with the Poincaré symmetry, by Noether's theorem, imply 10 conservation laws: 1 for the energy – associated with translations...

Chapter V Hall 2015, Theorem 7.35 Humphreys 1972, Section 16 Humphreys 1972, Part (b) of Theorem 18.4 Humphreys 1972 Section 18.3 and Theorem 18.4 Conway, John;...

weaker condition (2) is actually equivalent to (1), as stated by Engel's theorem: A finite dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is...

Compact group Simple Lie group Borel subalgebra Jacobson–Morozov theorem Serre 2000, Ch. II, § 2, Corollary to Theorem 3. Since the Killing form B is...

laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a...

for a partial list of real simple Lie algebras. Fulton & Harris 1991, Theorem 9.26. Fulton & Harris 1991, § 21.1. Fulton & Harris 1991, § 21.2. Simple...