Abstract For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A ( h ) , called the universal algebra of h , as a quotient of the polynomial algebra k [ X i j | i , j = 1 , ⋯ , n ] through an ideal generated by n 3 polynomials. We prove that A ( h ) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h . The new object A ( h ) is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group Aut L b z ( h ) of h is isomorphic to the group U ( G ( A ( h ) o ) ) of all invertible group-like elements of the finite dual A ( h ) o . Secondly, for an abelian group G, we show that there exists a bijection between the set of all G-gradings on h and the set of all bialgebra homomorphisms A ( h ) → k [ G ] . Based on this, all G-gradings on h are explicitly classified and parameterized. A ( h ) is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra h .

中文翻译：

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有限维莱布尼茨/李代数的一个新不变量

摘要 对于域 k 上的 n 维莱布尼茨/李代数 h 我们引入了一个新的不变量 A ( h )，称为 h 的泛代数，作为多项式代数 k [ X ij | 的商] i , j = 1 , ⋯ , n ] 通过由 n 3 个多项式生成的理想。我们证明了 A ( h ) 承认一个独特的双代数结构，这使它成为所有作用于 h 的可交换双代数中的初始对象。新对象 A ( h ) 是回答李代数理论中两个开放问题的关键工具。首先，我们证明 h 的自同构群 Aut L bz ( h ) 与有限对偶 A ( h ) o 的所有类可逆群元素的群 U ( G ( A ( h ) o ) ) 同构。其次，对于阿贝尔群 G，我们证明 h 上所有 G 分级的集合与所有双代数同态 A ( h ) → k [ G ] 的集合之间存在双射。基于此，h 上的所有 G 分级都被明确分类和参数化。A ( h ) 还用于证明存在与任何有限维莱布尼茨代数 h 相关联的通用交换 Hopf 代数。