论文标题
Leibniz代数及其内部自动形态的对称多项式
Symmetric polynomials in Leibniz algebras and their inner automorphisms
论文作者
论文摘要
令$ l_n $为$ x_n = \ {x_1，\ ldots，x_n \} $在特征零的字段$ k $上生成的免费metabelian leibniz代数。这是$ 2 $ 2 $ leibniz代数的各种可解决方案中的排名$ n $的免费代数。如果$ s（x_ {σ（1）}，\ ldots，x_ {σ（n）}）= s（x_1，\ ldots，x_n）$ in l_n $ symmetric中的元素$ s（x_n）\ in l_n $对称。 $ l_n $的对称多项式的$ l_n^{s_n} $是对称组的代数$ s_n $。令$ k [x_n] $为通常的多项式代数，而$ x_n $不确定。代数$ k [x_n]^{s_n} $的描述是众所周知的，而换向器中的代数$（l_n'）^{s_n} $在commutator poldumutator bexple $ l_n'$中是一个正确的$ k [x_n]^{x_n]^{s_n} $ - module。我们给出$ k [x_n]^{s_n} $ - 模块$（l_n'）^{s_n} $的明确形式的元素。此外，我们确定组的$ {\ rm inn}的描述（l_ {n}^{s_n}）$的内部自动形态$ l_n^{s_n} $。这些发现可以被视为对自由Metabelian Lie代数获得的最新结果的概括。
Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. This is the free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz algebras. We call an element $s(X_n)\in L_n$ symmetric if $s(x_{σ(1)},\ldots,x_{σ(n)})=s(x_1,\ldots,x_n)$ for each permutation $σ$ of $\{1,\ldots,n\}$. The set $L_n^{S_n}$ of symmetric polynomials of $L_n$ is the algebra of invariants of the symmetric group $S_n$. Let $K[X_n]$ be the usual polynomial algebra with indeterminates from $X_n$. The description of the algebra $K[X_n]^{S_n}$ is well known, and the algebra $(L_n')^{S_n}$ in the commutator ideal $L_n'$ is a right $K[X_n]^{S_n}$-module. We give explicit forms of elements of the $K[X_n]^{S_n}$-module $(L_n')^{S_n}$. Additionally, we determine the description of the group ${\rm Inn}(L_{n}^{S_n})$ of inner automorphisms of the algebra $L_n^{S_n}$. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank $n$.
