|
In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal ''G''-bundle ''P'' over a manifold ''M'', where ''G'' is a Lie group, is the Lie algebroid of the gauge groupoid of ''P''. Explicitly, it is given by the following short exact sequence of vector bundles over ''M'': : It is named after Michael Atiyah, who introduced the construction to study the existence theory of complex analytic connections, and it has applications in gauge theory and mechanics. ==Direct construction== For any fiber bundle ''P'' over a manifold ''M'', with projection ''π'': ''P''→''M'', the differential d''π'' of ''π'' defines a short exact sequence : of vector bundles over ''P'', where the vertical bundle ''VP'' is the kernel of the differential projection. If ''P'' is a principal ''G''-bundle, then the group ''G'' acts on the vector bundles in this sequence. The vertical bundle is isomorphic to the trivial g bundle over ''P'', where g is the Lie algebra of ''G'', and the quotient by the diagonal ''G'' action is the associated bundle ''P'' ×''G'' g. The quotient by ''G'' of this exact sequence thus yields the Atiyah sequence of vector bundles over ''M''. 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「Atiyah algebroid」の詳細全文を読む スポンサード リンク
|