伴随丛

在数学中，伴随丛（adjoint bundle）是一个自然相配于任何主丛的向量丛。伴随丛的纤维带有李代数结构使得伴随丛从未一个代数丛。伴随丛在联络理论以及正规理论中具有重要的应用。

形式定义

Let G be a Lie group with Lie algebra ${\mathfrak {g}}$, and let P be a principal G-bundle over a smooth manifold M. Let

$\mathrm {Ad} :G\to \mathrm {Aut} ({\mathfrak {g}})\subset \mathrm {GL} ({\mathfrak {g}})$

be the adjoint representation of G. The adjoint bundle of P is the associated bundle

$\mathrm {Ad} _{P}=P\times _{\mathrm {Ad} }{\mathfrak {g}}$

The adjoint bundle is also commonly denoted by ${\mathfrak {g}}_{P}$. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, x] for p ∈ P and x ∈ ${\mathfrak {g}}$ such that

$[p\cdot g,x]=[p,\mathrm {Ad} _{g}(x)]$

for all g ∈ G. Since the structure group of the adjoint bundle consists of Lie algebra automorphisms, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over M.

性质

Differential forms on M with values in Ad_P are in one-to-one corresponding with horizontal, G-equivariant Lie algebra-valued forms on P. A prime example is the curvature of any connection on P which may be regarded as a 2-form on M with values in Ad_P.

The space of sections of the adjoint bundle is naturally an (infinite-dimensional) Lie algebra. It may be regarded as the Lie algebra of the infinite-dimensional Lie group of gauge transformations of P which can be thought of as sections of the bundle P ×_Ψ G where Ψ is the action of G on itself by conjugation.