论文标题
对称多项式的谎言代数的内部自动形态
Inner automorphisms of Lie algebras of symmetric polynomials
论文作者
论文摘要
令$ l_ {n} $为免费的lie代数，$ f_ {n} $是免费的metabelian lie代数，$ l_ {n，c} $是免费的$ c $ c $ c $ c $ c $ lie代数等级$ n $ n $的免费metabelian nilpotent，由$ x_1，x_1，x_1 $ x_ $ a $ ken $ ken $ k $ ken $ k $ ken $ k $ ken。如果$ p（x_1，\ ldots，x_n）= p（x_ {π（1）}，\ ldots，x_ {π（n））$，我们将每个元素的$ p（x_1，\ ldots，x_n）= p（x_1，\ ldots，x_n），x_ {π（n））$在这些元素中称为多项式$ p（x_n）$对称。集合$ l_n^{s_n} $，$ f_n^{s_n} $，以及$ l_ {n，c}^{s_n} $对称多项式的$与集团$ s_n $ in $ s_n $的代数相吻合， 分别。我们确定$ \ text {inn}（f_ {n}^{s_n}）$和$ \ text {inn}（l_ {n，c}^{s_n}）$的内部自动形态的$ f_ {n}^n}^n}^n}^{s_n} $ n} $ f_ {特别是，我们获得了组的描述$ \ text {aut}（l_ {2}^{s_2}）$，$ \ \ \ text {aut}（f_ {2}^{s_2}）$ $ l_ {2}^{s_2} $，$ f_ {2}^{s_2} $和$ l_ {2，c}^{s_2} $。
Let $L_{n}$ be the free Lie algebra, $F_{n}$ be the free metabelian Lie algebra, and $L_{n,c}$ be the free metabelian nilpotent of class $c$ Lie algebra of rank $n$ generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ symmetric in these Lie algebras if $p(x_1,\ldots,x_n)=p(x_{π(1)},\ldots,x_{π(n)})$ for each element $π$ of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincide with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\text{Inn}(F_{n}^{S_n})$ and $\text{Inn}(L_{n,c}^{S_n})$ of inner automorphisms of the algebras $F_{n}^{S_n}$ and $L_{n,c}^{S_n}$, respectively. In particular, we obtain the descriptions of the groups $\text{Aut}(L_{2}^{S_2})$, $\text{Aut}(F_{2}^{S_2})$, and $\text{Aut}(L_{2,c}^{S_2})$ of all automorphisms of the algebras $L_{2}^{S_2}$, $F_{2}^{S_2}$, and $L_{2,c}^{S_2}$, respectively.
