Associative, Lie, and left-symmetric algebras of derivations

Abstract

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric algebra of all derivations of $P_n$ \cite{Dzhuma99,UU2014-1}. Using the language of $\mathscr{L}_n$, for every derivation $D\in \mathscr{L}_n$ we define the associative algebra $A_D$, the Lie algebra $L_D$, and the left-symmetric algebra $\mathscr{L}_D$ related to the study of the Jacobian Conjecture. For every derivation $D\in \mathscr{L}_n$ there is a unique $n$-tuple $F=(f_1,f_2,\ldots,f_n)$ of elements of $P_n$ such that $D=D_F=f_1\partial_1+f_2\partial_2+\ldots+f_n\partial_n$. In this case, using an action of the Hopf algebra of noncommutative symmetric functions $\mathrm{NSymm}$ on $P_n$, we show that these algebras are closely related to the description of coefficients of the formal inverse to the polynomial endomorphism $X+tF$, where $X=(x_1,x_2,\ldots,x_n)$ and $t$ is an independent parameter. We prove that the Jacobian matrix $J(F)$ is nilpotent if and only if all right powers $D_F^{[r]}$ of $D_F$ in $\mathscr{L}_n$ have zero divergence. In particular, if $J(F)$ is nilpotent then $D_F$ is right nilpotent. We discuss some advantages and shortcomings of these algebras and formulate some open questions.