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Lie algebra-valued form : ウィキペディア英語版
Lie algebra-valued differential form
In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.
== Wedge product ==

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation, denoted by (), is given by: for \mathfrak-valued ''p''-form \omega and \mathfrak-valued ''q''-form \eta
:()(v_1, \cdots, v_) = \sum_ \operatorname(\sigma) (\cdots, v_), \eta(v_, \cdots, v_) )
where ''v''''i'''s are tangent vectors. The notation is meant to indicate both operations involved. For example, if \omega and \eta are Lie algebra-valued one forms, then one has
:()(v_1,v_2) = (() - ()).

The operation () can also be defined as the bilinear operation on \Omega(M, \mathfrak g) satisfying
:(\otimes \alpha) \wedge (h \otimes \beta) ) = (h ) \otimes (\alpha \wedge \beta)
for all g, h \in \mathfrak g and \alpha, \beta \in \Omega(M, \mathbb R).
Some authors have used the notation (\eta ) instead of (). The notation (\eta ), which resembles a commutator, is justified by the fact that if the Lie algebra \mathfrak g is a matrix algebra then () is nothing but the graded commutator of \omega and \eta, i. e. if \omega \in \Omega^p(M, \mathfrak g) and \eta \in \Omega^q(M, \mathfrak g) then
:() = \omega\wedge\eta - (-1)^\eta\wedge\omega,
where \omega \wedge \eta,\ \eta \wedge \omega \in \Omega^(M, \mathfrak g) are wedge products formed using the matrix multiplication on \mathfrak g.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Lie algebra-valued differential form」の詳細全文を読む



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