In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra...

In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations...

with any tensorial forms on P {\displaystyle P} of adjoint type. Maurer–Cartan form Adjoint bundle S. Kobayashi, K. Nomizu. Foundations of Differential...

The Lie algebra adjoint bundle is usually denoted ad ⁡ ( P ) {\displaystyle \operatorname {ad} (P)} , and the Lie group adjoint bundle by Ad ⁡ ( P ) {\displaystyle...

connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are D A ∗ F A + [ Φ , D A Φ ] = 0 , D...

field on the spacetime with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see vector bundle for necessary definitions). Throughout...

(mathematics), a connection over a principal bundle G with a section (the Higgs field) of the associated adjoint bundle Monopole, the first term in a multipole...

this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics...

section of the vertical bundle V of P. Hence it is basic and so is determined by a 1-form on M with values in the adjoint bundle g P := P × G g . {\displaystyle...

{C} )} there is still a natural associated vector bundle to P {\displaystyle P} , the adjoint bundle ad ⁡ P {\displaystyle \operatorname {ad} P} , with...

mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. Physically, the section can be interpreted as...

{\displaystyle G} -bundle over a 3-manifold M {\displaystyle M} , Φ {\displaystyle \Phi } is a section of the corresponding adjoint bundle, d A {\displaystyle...

difference of two adapted connections is a 1-form on M with values in the adjoint bundle AdQ. That is to say, the space AQ of adapted connections is an affine...

capital A adjoint bundle Ad ⁡ ( F ( E ) ) {\displaystyle \operatorname {Ad} ({\mathcal {F}}(E))} of the frame bundle of the vector bundle E {\displaystyle...

self-adjoint operator is an operator equal to its own (formal) adjoint. If Ω is a domain in Rn, and P a differential operator on Ω, then the adjoint of...

bundle. This connection has a curvature form F A {\displaystyle F_{A}} , which is a two-form on X {\displaystyle X} with values in the adjoint bundle...

resulting in an isomorphism between the transpose and adjoint of u. The matrix of the adjoint of a map is the transposed matrix only if the bases are...

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

1-forms, where ad ( P ) {\displaystyle {\text{ad}}(P)} is the associated adjoint bundle. For any non-empty subset X of an affine space A, there is a smallest...

MR 1207013. Fujita, Takao (1987), "On polarized manifolds whose adjoint bundles are not semipositive", Algebraic geometry, Sendai, 1985, Adv. Stud...

vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B...

Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle. Algebra bundle Adjoint bundle A. Weinstein...

are implemented for G-invariant self-adjoint operators between vector bundles associated to some principal G-bundle, although the precise conditions under...

section through that bundle. Coupled to the frame bundle is a second bundle, the spinor bundle. A section through the spinor bundle is just the particle...

G} , its quotient by the diagonal G {\displaystyle G} action is the adjoint bundle P × G g {\displaystyle P\times _{G}{\mathfrak {g}}} . In conclusion...

representable if and only if it has a left adjoint. The categorical notions of universal morphisms and adjoint functors can both be expressed using representable...

linear operator, and is known as the transpose or composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer...

flat G-principal bundle P on M there exists a unique homomorphism, called the Chern–Weil homomorphism, from the algebra of G-adjoint invariant polynomials...

on P {\displaystyle P} , its isotropy Lie algebra bundle is isomorphic to the adjoint vector bundle P × G g {\displaystyle P\times _{G}{\mathfrak {g}}}...

group bundle P ~ → X {\displaystyle {\widetilde {P}}\to X} whose typical fiber is a group G {\displaystyle G} which acts on itself by the adjoint representation...