论文标题
有限尺寸莱布尼兹/谎言代数的新不变
A new invariant for finite dimensional Leibniz/Lie algebras
论文作者
论文摘要
对于$ n $维的leibniz/lie代数$ \ mathfrak {h} $在字段$ k $上，我们介绍了一个新的不变$ {\ mathcal a}（\ mathfrak {h {h}）$，称为\ emph {noversal algebra} $ $ k [x_ {ij} \，| \，i，j = 1，\ cdots，n] $通过$ n^3 $多项式生成的理想。我们证明$ {\ mathcal a}（\ mathfrak {h}）$允许独特的bialgebra结构，这使其成为所有交换性的bialgebras在$ \ mathfrak {h h} $上均方根的初始对象。新对象$ {\ mathcal a}（\ mathfrak {h}）$是回答谎言代数理论的两个开放问题的关键工具。首先，我们证明了$ \ Mathfrak {h} $ of $ \ mathfrak {h} $的自动形态组$ {\ rm aut} _ {\ mathfrak {h}）$是$ \ u \ bigl（g（g（{{\ mathcal a}（\ mathfrak），有限双$ {\ mathcal a}（\ mathfrak {h}）^{\ rm o} $的有限双元素的类似群的元素。其次，对于Abelian Group $ G $，我们表明在$ \ Mathfrak {H} $上的所有$ G $ - 级别与所有Bialgebra同型同构$ {\ Mathcal a}（\ Mathfrak {H}）\ k [g [g] $之间都存在两者。基于此，明确对$ \ mathfrak {h} $上的所有$ g $ - 级别进行了明确的分类和参数化。 $ {\ mathcal a}（\ mathfrak {h}）$也用于证明存在与任何有限尺寸leibniz代数$ \ mathfrak {h h} $相关的通用交换hopf代数。
For an $n$-dimensional Leibniz/Lie algebra $\mathfrak{h}$ over a field $k$ we introduce a new invariant ${\mathcal A}(\mathfrak{h})$, called the \emph{universal algebra} of $\mathfrak{h}$, as a quotient of the polynomial algebra $k[X_{ij} \, | \, i, j = 1, \cdots, n]$ through an ideal generated by $n^3$ polynomials. We prove that ${\mathcal A}(\mathfrak{h})$ admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on $\mathfrak{h}$. The new object ${\mathcal A} (\mathfrak{h})$ is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group ${\rm Aut}_{Lbz} (\mathfrak{h})$ of $\mathfrak{h}$ is isomorphic to the group $U \bigl( G({\mathcal A} (\mathfrak{h})^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual ${\mathcal A} (\mathfrak{h})^{\rm o}$. Secondly, for an abelian group $G$, we show that there exists a bijection between the set of all $G$-gradings on $\mathfrak{h}$ and the set of all bialgebra homomorphisms ${\mathcal A} (\mathfrak{h}) \to k[G]$. Based on this, all $G$-gradings on $\mathfrak{h}$ are explicitly classified and parameterized. ${\mathcal A} (\mathfrak{h})$ is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra $\mathfrak{h}$.
