ON AUTOMORPHISMS OF LIE ALGEBRA OF SYMMETRIC POLYNOMIALS

Öz

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.

Anahtar Kelimeler

• Lie algebras
• metabelian
• nilpotent
• symmetric polynomials
• automorphisms

Kaynakça

1. Yu.A. Bahturin, Identical Relations in Lie Algebras (Russian), Nauka, Moscow, (1985). Translation: VNU Science Press, Utrecht, (1987).
2. P.M. Cohn, Subalgebras of free associative algebras, Proc. London Math. Soc. 14(3), pp. 618-632 (1964).
3. V. Drensky, Fixed algebras of residually nilpotent Lie algebras, Proc. Amer. Math. Soc. 120(4), pp. 1021-1028 (1994).
4. V. Drensky, {\c S}. F{\i}nd{\i}k, N. {\c S}. \"O{\u g}\"u{\c s}l\"u, Symmetric polynomials in the free metabelian Lie algebras, Mediterranean Journal of Mathematics 17(5), pp. 1-11 (2020).
5. {\c S}. F{\i}nd{\i}k, N. {\c S}. \"O{\u g}\"u{\c s}l\"u, Palindromes in the free metabelian Lie algebras, Internat. J. Algebra Comput. 29(5), pp. 885-891 (2019).
6. M.C. Wolf, Symmetric functions of non-commutative elements, Duke Math. J. 2(4), pp. 626-637 (1936).