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

the Weyl 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...

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...

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

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

conjectured, and Schmid (1976) proved, a geometric analogue of the Borel–Bott–Weil theorem, for the discrete series, using L2 cohomology instead of the coherent...

q = 4, CII with p = 1 or q = 1, EII, EVI, EIX, FI and G. In the Bott periodicity theorem, the loop spaces of the stable orthogonal group can be interpreted...

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

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

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...

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

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...

The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish...

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;...

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

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...

Frankel 1959 Milnor 1963, p. 39 Bott 1959 Lazarsfeld 2004, Example 3.1.24 Voisin 2003, Theorem 1.29 Lazarsfeld 2004, Theorem 3.1.13 Lazarsfeld 2004, Example...

Bass-Serre theorem (group theory) Borel–Bott–Weil theorem (representation theory) Borel–Weil theorem (representation theory) Brauer–Nesbitt theorem (representation...

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...

analog of Schur's lemma Hall 2015 Theorem 5.6 Hall 2013 Section 17.3 Hall 2015 Theorem 4.29 Dixmier 1977, Theorem 1.6.3 Hall 2015 Section 4.3 Hall 2015...

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...

h+\mathbb {C} e} -weight vector ( b {\displaystyle {\mathfrak {b}}} is a Borel subalgebra). Then Those v j {\displaystyle v_{j}} 's that are nonzero are...

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...