nLab
equivariant connection
Contents
Context
$\infty$ChernWeil theory
Differential cohomology
differential cohomology
Ingredients
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Fiber integration
Application to gauge theory
Representation theory
representation theory
geometric representation theory
Ingredients
Definitions
representation, 2representation, ∞representation

group, ∞group

group algebra, algebraic group, Lie algebra

vector space, nvector space

affine space, symplectic vector space

action, ∞action

module, equivariant object

bimodule, Morita equivalence

induced representation, Frobenius reciprocity

Hilbert space, Banach space, Fourier transform, functional analysis

orbit, coadjoint orbit, Killing form

unitary representation

geometric quantization, coherent state

socle, quiver

module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory

Dmodule, perverse sheaf,

Grothendieck group, lambdaring, symmetric function, formal group

principal bundle, torsor, vector bundle, Atiyah Lie algebroid

geometric function theory, groupoidification

EilenbergMoore category, algebra over an operad, actegory, crossed module

reconstruction theorems
Theorems
Contents
Idea
An equivariant connection is a connection on a bundle $\nabla$ over a space $X$ with action by a group $H$ which is equipped with $H$equivariant structure, hence equivalently – in the language of higher differential geometry of smooth groupoids – an extension of a connection $\nabla \;\colon\; X \longrightarrow \mathbf{B}G_{conn}$ to a connection $\nabla_{equ}$ on the action groupoid $X//H$:
$\array{
X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}G_{conn}
\\
\downarrow & \nearrow_{\mathrlap{\nabla_{equ}}}
\\
X//H
}
\,.$
Examples
Last revised on March 15, 2021 at 03:31:50.
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