m-commuting maps on triangular and strictly triangular infinite matrices

Let $N_\infty(F)$ be the ring of infinite strictly upper triangular matrices with entries in an infinite field. The description of the commuting maps defined on $N_\infty(F)$, i.e. the maps $f\colon N_\infty(F)\rightarrow N_\infty(F)$ such that $[f(X),X]=0$ for every $X\in N_\infty(F)$, is presented. With the use of this result, the form of $m$-commuting maps defined on $T_\infty(F)$ -- the ring of infinite upper triangular matrices, i.e. the maps $f\colon T_\infty(F)\rightarrow T_\infty(F)$ such that $[f(X),X^m]=0$ for every $X\in T_\infty(F)$, is found.

n-Derivations of the Extended Schrödinger–Virasoro Lie Algebra

Let [Formula: see text] be the extended Schrödinger–Virasoro Lie algebra and [Formula: see text] an integer. A map [Formula: see text] is called an [Formula: see text]-derivation if it is a derivation in one variable while other variables fixed. We investigate [Formula: see text]-derivations of the extended Schrödinger–Virasoro Lie algebra [Formula: see text]. The main result when [Formula: see text] is then applied to characterize the linear commuting maps and the commutative post-Lie algebra structures on [Formula: see text].