ON AUTOMORPHISMS OF LIE ALGEBRA OF SYMMETRIC POLYNOMIALS

Authors : Şehmus FINDIK, Nazar Şahin ÖĞÜŞLÜ

Pages : 49-54

Doi:10.33773/jum.1165977

View : 21 | Download : 12

Publication Date : 2023-01-31

Article Type : Research Paper

Abstract :Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ of characteristic zero, $L_{n,c}=L_{n}/insert ignore into journalissuearticles values(L_{n}\`\`+\\gamma_{c+1}insert ignore into journalissuearticles values(L_{n}););$ 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 $pinsert ignore into journalissuearticles values(X_n);$ in these Lie algebras {\\it symmetric} if $pinsert ignore into journalissuearticles values(x_1,\\ldots,x_n);=pinsert ignore into journalissuearticles values(x_{\\piinsert ignore into journalissuearticles values(1);},\\ldots,x_{\\piinsert ignore into journalissuearticles values(n);});$ 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 $\\text{\\rm Inn}insert ignore into journalissuearticles values(L_{n,c}^{S_n});\\cap \\text{\\rm Inn}insert ignore into journalissuearticles values(L_{n,c});$ and $\\text{\\rm Inn}insert ignore into journalissuearticles values(F_{n}^{S_n});\\cap \\text{\\rm Inn}insert ignore into journalissuearticles values(F_{n});$ of inner automorphisms of the algebras $L_{n,c}^{S_n}$ and $F_{n}^{S_n}$ in the groups $\\text{\\rm Inn}insert ignore into journalissuearticles values(L_{n,c});$ and $\\text{\\rm Inn}insert ignore into journalissuearticles values(F_{n});$, respectively. In particular, we obtain the descriptions of the groups $\\text{\\rm Aut}insert ignore into journalissuearticles values(L_{2}^{S_2});\\cap \\text{\\rm Aut}insert ignore into journalissuearticles values(L_{2});$ and $\\text{\\rm Aut}insert ignore into journalissuearticles values(F_{2}^{S_2});\\cap \\text{\\rm Aut}insert ignore into journalissuearticles values(F_{2});$ of automorphisms of the algebras $L_{2}^{S_2}$ and $F_{2}^{S_2}$ in the groups $\\text{\\rm Aut}insert ignore into journalissuearticles values(L_{2});$ and $\\text{\\rm Aut}insert ignore into journalissuearticles values(F_{2});$, respectively.
Keywords : Lie algebras, metabelian, nilpotent, symmetric polynomials, automorphisms