非退化双线性形式与左对称结构
Non-degenerate bilinear forms and left-symmetric structures
AI总结：
本文研究非退化双线性形式诱导左对称结构的充要条件，证明其完备性等价于李代数幺模性，并引入双扩张构造一类非退化双线性形式。
AI中文摘要：
Chu证明了偶数维李代数上的辛结构$\omega$诱导一个左对称结构，该结构由$\omega(x\scriptstyle{\Delta}_{\omega} y,z)=-\omega(y,[x,z])$定义。El Bourkadi和Mansouri证明了奇数维李代数$\mathfrak{g}$上的余辛结构诱导左对称结构，该结构利用与余辛结构相关的从$\mathfrak{g}$到$\mathfrak{g}^*$的线性同构来定义。本文中，对于李代数$\mathfrak{g}$上的非退化双线性形式$\phi$，我们给出了由$\phi(x\scriptstyle{\Delta}_{\phi} y,z)=-\phi(y,[x,z])$定义的乘积$\scriptstyle{\Delta}_{\phi}$成为左对称结构的充分必要条件。我们还证明了左对称结构$\scriptstyle{\Delta}_{\phi}$是完备的当且仅当李代数$\mathfrak{g}$是幺模的。此外，我们提出了非退化双线性形式的双扩张概念，并证明了一类特定的非退化双线性形式可通过双扩张得到。
英文摘要：
Chu proved that a symplectic structure $ω$ on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by $ω(x\scriptstyleΔ_ω y,z)=-ω(y,[x,z])$. El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra $\mathfrak{g}$ induces the left-symmetric structure, which is defined by using a linear isomorphism from $\mathfrak{g}$ to $\mathfrak{g}^*$ associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form $ϕ$ on a Lie algebra $\mathfrak{g}$, we give a necessary and sufficient condition for the product $\scriptstyleΔ_ϕ$ on $\mathfrak{g}$ defined by $ϕ(x\scriptstyleΔ_ϕ y,z)=-ϕ(y,[x,z])$ to be a left-symmetric structure. We also prove that the left-symmetric structure $\scriptstyleΔ_ϕ$ is complete if and only if the Lie algebra $\mathfrak{g}$ is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.
