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searching for Complexification (Lie group) 36 found (108 total)

alternate case: complexification (Lie group)

Zonal spherical function (6,698 words) [view diff] no match in snippet view article find links to article

Harish-Chandra's formula. The group G = SL(2,C) is the complexification of the compact Lie group K = SU(2) and the double cover of the Lorentz group. The

Cayley plane (412 words) [view diff] no match in snippet view article find links to article

and 16. The complex Cayley plane is a homogeneous space under the complexification of the group E6 by a parabolic subgroup P1. It is the closed orbit

Representation theory of SL2(R) (1,827 words) [view diff] no match in snippet view article

(1947), and Harish-Chandra (1952). We choose a basis H, X, Y for the complexification of the Lie algebra of SL(2, R) so that iH generates the Lie algebra

Spin group (3,989 words) [view diff] no match in snippet view article find links to article

In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n

Theorem of the highest weight (1,103 words) [view diff] no match in snippet view article find links to article

theorem classifying the irreducible representations of a connected compact Lie group K {\displaystyle K} . The theorem states that there is a bijection λ ↦

Iwasawa decomposition (990 words) [view diff] no match in snippet view article find links to article

semisimple real Lie group. g 0 {\displaystyle {\mathfrak {g}}_{0}} is the Lie algebra of G g {\displaystyle {\mathfrak {g}}} is the complexification of g 0 {\displaystyle

Harish-Chandra class (200 words) [view diff] no match in snippet view article find links to article

an element of the connected component of the Lie group of Lie algebra automorphisms of the complexification g⊗C. The subgroup Gss of G generated by the

Cartan decomposition (1,499 words) [view diff] no match in snippet view article find links to article

the Lie algebra of a compact semisimple Lie group. Let g {\displaystyle {\mathfrak {g}}} be the complexification of a real semisimple Lie algebra g 0 {\displaystyle

Eisenstein integral (470 words) [view diff] no match in snippet view article find links to article

= MAN is a cuspidal parabolic subgroup of G ν is an element of the complexification of a a is the Lie algebra of A in the Langlands decomposition P = MAN

Maximal torus (1,734 words) [view diff] no match in snippet view article find links to article

subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic

Lie group–Lie algebra correspondence (4,466 words) [view diff] no match in snippet view article find links to article

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for

Compact group (4,472 words) [view diff] no match in snippet view article find links to article

Lie algebras. Indeed, if K is a simply connected compact Lie group, then the complexification of the Lie algebra of K is semisimple. Conversely, every

Complex Lie algebra (830 words) [view diff] no match in snippet view article find links to article

said to be a real form of g {\displaystyle {\mathfrak {g}}} if the complexification g 0 ⊗ R C {\displaystyle {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb

Biquaternion (3,365 words) [view diff] no match in snippet view article find links to article

(real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions

Harmonic superspace (964 words) [view diff] no match in snippet view article find links to article

turns out that the 8 real SUSY generators are pseudoreal, and after complexification, correspond to the tensor product of a four-dimensional Dirac spinor

Quantum group (4,983 words) [view diff] no match in snippet view article find links to article

starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain

Kostant's convexity theorem (1,901 words) [view diff] no match in snippet view article find links to article

semisimple Lie group G. The result discussed above for compact Lie groups K corresponds to the special case when G is the complexification of K: in this

Semi-simplicity (1,867 words) [view diff] no match in snippet view article find links to article

{\displaystyle {\mathfrak {g}}} is the complexification of the Lie algebra of a simply connected compact Lie group K {\displaystyle K} . Since K {\displaystyle

Euler class (2,003 words) [view diff] no match in snippet view article find links to article

class of the complexification of E: p r ( E ) = c r ( C ⊗ E ) {\displaystyle p_{r}(E)=c_{r}(\mathbf {C} \otimes E)} . The complexification C ⊗ E {\displaystyle

Plancherel theorem for spherical functions (11,769 words) [view diff] no match in snippet view article find links to article

is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup U, a compact semisimple Lie group. If g {\displaystyle {\mathfrak

Symplectic vector space (2,275 words) [view diff] no match in snippet view article find links to article

analog to a Lagrangian subspace is a real subspace, a subspace whose complexification is the whole space: W = V ⊕ J V. As can be seen from the standard symplectic

Representation theory of SU(2) (3,369 words) [view diff] no match in snippet view article

of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition

Hitchin's equations (1,350 words) [view diff] no match in snippet view article find links to article

{\text{ad}}P^{\mathbb {C} }} is the complexification of the adjoint bundle of P {\displaystyle P} , with fibre given by the complexification g ⊗ C {\displaystyle {\mathfrak

Segal–Bargmann space (2,026 words) [view diff] no match in snippet view article find links to article

Lie group, such as SU(N). The role of the phase space C n {\displaystyle \mathbb {C} ^{n}} is then played by the complexification of the compact Lie group

Butcher group (4,050 words) [view diff] no match in snippet view article find links to article

John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear

Kazhdan–Lusztig polynomial (3,330 words) [view diff] no match in snippet view article find links to article

of a Coxeter group W, which can in particular be the Weyl group of a Lie group. In the spring of 1978 Kazhdan and Lusztig were studying Springer representations

Prehomogeneous vector space (1,712 words) [view diff] no match in snippet view article find links to article

of view of Lie theory: for instance, in Knapp (2002), G is a complex Lie group and V is a holomorphic representation of G with an open dense orbit. The

Lorentz invariance in loop quantum gravity (1,300 words) [view diff] no match in snippet view article find links to article

Immirzi parameter is necessary to resolve an ambiguity in the process of complexification. These are some of the many ways in which different quantizations of

Chern–Weil homomorphism (2,782 words) [view diff] no match in snippet view article find links to article

step in the theory of characteristic classes. Let G be a real or complex Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} , and let C [ g ] {\displaystyle

Kostant polynomial (2,075 words) [view diff] no match in snippet view article find links to article

generalized flag manifold K/T, also isomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borel showed

Freudenthal magic square (2,914 words) [view diff] no match in snippet view article find links to article

automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector

Hurwitz's theorem (composition algebras) (3,682 words) [view diff] no match in snippet view article

orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of A, an N-dimensional complex space. If N is even, N

Littelmann path model (2,147 words) [view diff] no match in snippet view article find links to article

{\displaystyle {\mathfrak {g}}} is the complexification of the Lie algebra of a compact connected simply connected semisimple Lie group. The subalgebra g 1 {\displaystyle

Weyl's theorem on complete reducibility (2,465 words) [view diff] no match in snippet view article find links to article

{\displaystyle {\mathfrak {g}}} is the complexification of the Lie algebra of a simply connected compact Lie group K {\displaystyle K} . (If, for example

Clebsch–Gordan coefficients for SU(3) (7,750 words) [view diff] no match in snippet view article

factors of i in these formulas, this is technically a basis for the complexification of the su(3) Lie algebra, namely sl(3,C). The preceding basis is then

Schwarzian derivative (7,160 words) [view diff] no match in snippet view article find links to article

+ n . {\displaystyle [d_{m},d_{n}]=(m-n)d_{m+n}.} A basis for the complexification of Fλ(S1) is given by v n = e i n θ ( d θ ) λ , {\displaystyle v_{n}=e^{in\theta