论文标题
平面符号谎言代数
Flat symplectic Lie algebras
论文作者
论文摘要
令$（g，ω）$为一个符号谎言组，即，一个具有左不变符号形式的谎言组。如果$ \ g $是$ g $的lie代数，那么我们称$（\ g，ω= \ om（e））$ sympletic lie lie代数。 $ \ g $上的产品$ \ bullet $由$3Ω\ left（x \ bullet y，z \ right）=ω\ left（[x，y]，z \ prirs）+ω\ left（[x，z，z]，y \ first）$扩展到左na $ g $ g $ na $ g $ na na is pplection $ na na $ na $ na $ o.当$ \ na $具有消失的曲率时，我们将$（g，ω）$称为平坦的simpletic Lie组和$（\ g，\ om）$ a flat Symbletic Lie代数。在本文中，我们研究了扁平的同胞谎言组。首先，我们表明，相对于$ \ om $，平面符号谎言代数的派生理想是退化的。我们表明，一个平坦的符号谎言组必须与退化中心保持敏捷。这意味着一个平面符号谎言组的连接$ \ na $始终完成。我们证明可以应用双重扩展过程来表征所有平面符号谎言代数。更确切地说，我们表明，每个平面符号谎言代数都是通过$ \ {0 \} $开始的平面符号谎言代数的一系列双重扩展而获得的。作为低维度的示例，我们将所有平面符号谎言代数分类为$ \ leq6 $。
Let $(G,Ω)$ be a symplectic Lie group, i.e, a Lie group endowed with a left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call $(\G,ω=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined by $3ω\left(x\bullet y,z\right)=ω\left([x,y],z\right)+ω\left([x,z],y\right)$ extends to a left invariant connection $\na$ on $G$ which is torsion free and symplectic ($\na\Om=0)$. When $\na$ has vanishing curvature, we call $(G,Ω)$ a flat symplectic Lie group and $(\G,\om)$ a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to $\om$. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection $\na$ of a flat symplectic Lie group is always complete. We prove that the double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of double extension of flat symplectic Lie algebras starting from $\{0\}$. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension $\leq6$.
