The existence and classification of couplings between Lie algebra bundles and tangent bundles

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• 作者：Xiaoyu Li^a ; A.S. Mishchenko^a ; ^b ; ^{asmish-prof@yandex.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：55R ; 57R ; 58H
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：201
• 期：Complete
• 页码：291-308
• 全文大小：410 K

文摘

In this paper we investigate the existence and classification of couplings between Lie algebra bundles and tangent bundles. We first give a sufficient and necessary condition on the existence of coupling between Lie algebra bundle (LAB) and the tangent bundle; i.e., we define a new topology on the group $\mathbf{Aut}\mathfrak{g}$ of all automorphisms of the Lie algebra g$\mathfrak{g}$, denoted by baea200f88e5">$\mathbf{Aut}{\mathfrak{g}}^{\delta }$, and show that the tangent bundle TM can be coupled with the Lie algebra bundle L if and only if L   admits a local trivial structure with structural group endowed with such new topology. Then we show that there is a bijection between Coup_g(M)${\mathit{Coup}}_{\mathfrak{g}}(M)$ the set of all isomorphism classes of couplings (L,Ξ)$(L,\mathrm{\Xi })$ and the set ${\mathit{LAB}}_{\mathfrak{g}}^{\delta }(M)$ of isomorphism classes of g$\mathfrak{g}$-bundles with structural group baea200f88e5">$\mathbf{Aut}{\mathfrak{g}}^{\delta }$.