- Journal of Universal Mathematics
- Vol: 6 Issue: 1
- ON AUTOMORPHISMS OF LIE ALGEBRA OF SYMMETRIC POLYNOMIALS
ON AUTOMORPHISMS OF LIE ALGEBRA OF SYMMETRIC POLYNOMIALS
Authors : Şehmus Findik, Nazar Şahin Öğüşlü
Pages : 49-54
Doi:10.33773/jum.1165977
View : 11 | Download : 5
Publication Date : 2023-01-31
Article Type : Research
Abstract :Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ of characteristic zero, $L_{n,c}=L_{n}/(L_{n}\'\'+\\gamma_{c+1}(L_{n}))$ be the free metabelian nilpotent of class $c$ Lie algebra, and $F_{n}=L_{n}/L_{n}\'\'$ be the free metabelian Lie algebra generated by $x_1,\\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ in these Lie algebras {\\it symmetric} if $p(x_1,\\ldots,x_n)=p(x_{\\pi(1)},\\ldots,x_{\\pi(n)})$ for each element 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 coincides with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\\text{\\rm Inn}(L_{n,c}^{S_n})\\cap \\text{\\rm Inn}(L_{n,c})$ and $\\text{\\rm Inn}(F_{n}^{S_n})\\cap \\text{\\rm Inn}(F_{n})$ of inner automorphisms of the algebras $L_{n,c}^{S_n}$ and $F_{n}^{S_n}$ in the groups $\\text{\\rm Inn}(L_{n,c})$ and $\\text{\\rm Inn}(F_{n})$, respectively. In particular, we obtain the descriptions of the groups $\\text{\\rm Aut}(L_{2}^{S_2})\\cap \\text{\\rm Aut}(L_{2})$ and $\\text{\\rm Aut}(F_{2}^{S_2})\\cap \\text{\\rm Aut}(F_{2})$ of automorphisms of the algebras $L_{2}^{S_2}$ and $F_{2}^{S_2}$ in the groups $\\text{\\rm Aut}(L_{2})$ and $\\text{\\rm Aut}(F_{2})$, respectively.Keywords : Lie algebras, metabelian, nilpotent, symmetric polynomials, automorphisms