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140J). By a connection in Lie algebroid (10) we mean a splitting of Atiyah sequence (13), ie a mapping ^:TM —» A (22) such that p>^« idip«, or, equivalently, a subbundle Be A such that A = <{ HI*Z , in P. PROOF. The equality implies A H Iza . = v(Ray-)t-z ) [H iz * ztP, is a connection On the other hand, ^'HjVH»z "* TJTzM is a linear isomorphism, thus It remains to show the smoothness of the distribution H*.

N r1 Let A:TM + A be any connection in A; then F<>/\s a connection in A . A. A. " = l F « A ( X ) , 6 l , X€3f(K), <5"€Sec^. 2. A CLASSIFICATION OF LIE ALGEBRAS. are equivalent. Q LIE ALGEBRQ1DS WITH SEKlSIFiPLE ISOTRQPY Let Of be any bundle of scroisimple Lie algebras- on a manifold. 1. For any Z -connection V one 2-form in t£, there exists exactly fulfilling condition (1°) from prop. 1, identity (3°). fulfils the Bianchi PROOF. It is easy to check that X for v,weT M is a derivation of the Lie algebra <£f , R being the curva ture tensor of V .

N 4 . 1 . The elements S7 and &M fulfil the following as&tions (1°) RX y*-- C^M(X,Y),«5], X,Y£36(M), S"eSecq[, where R denotes the curvature tensor of V , ie ,O» *€Sec`
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