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<p>The aim of this note is to prove some cohomological vanishing theorems for non solvable affinelike Lie algebras, say ALLA. There are some relevant consequences of our vanishing theorems: <OrderedList> <ListItem> <ItemNumber>(1)</ItemNumber> <ItemContent> <p>Every real non solvable affinelike Lie algebra is formally nondegenerate in the sense of A. Weinstein, (Theorem <InternalRef RefID="FPar7">1</InternalRef>).</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(2)</ItemNumber> <ItemContent> <p>Let G be a non solvable Lie group whose Lie algebra is an affinelike Lie algebra (g,e). If the radical of [ker(ad(e)),ker(ad(e))] is commutative, then G admits a left invariant symplectic structure if and only if it has an open coadjoint orbit, (Theorem <InternalRef RefID="FPar8">2</InternalRef>).</p> </ItemContent> </ListItem> </OrderedList> In Section <InternalRef RefID="Sec8">6</InternalRef>, we use our vanishing theorems to supply an algebraic proof of the normal form theorem for Lie non solvable a-algebroids.</p>

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Non degeneracy of affinelike lie algebra

  • 期刊论文

  • 发表日期：

    2025年6月17日

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  • Michel Nguifo Boyom,
  • Stephane Puechmorel

摘要

The aim of this note is to prove some cohomological vanishing theorems for non solvable affinelike Lie algebras, say ALLA. There are some relevant consequences of our vanishing theorems: (1)

Every real non solvable affinelike Lie algebra is formally nondegenerate in the sense of A. Weinstein, (Theorem 1).

(2)

Let G be a non solvable Lie group whose Lie algebra is an affinelike Lie algebra (g,e). If the radical of [ker(ad(e)),ker(ad(e))] is commutative, then G admits a left invariant symplectic structure if and only if it has an open coadjoint orbit, (Theorem 2).

 In Section 6, we use our vanishing theorems to supply an algebraic proof of the normal form theorem for Lie non solvable a-algebroids.