Associative, Lie, and left-symmetric algebras of derivations

TL;DR

This paper explores the algebraic structures related to derivations of polynomial algebras, linking them to the Jacobian Conjecture, and investigates properties of associated algebras and their relation to nilpotency of the Jacobian matrix.

Contribution

It introduces new algebraic frameworks involving derivations and their associated algebras, connecting these to the Jacobian Conjecture and providing criteria for nilpotency.

Findings

The algebras are related to coefficients of formal inverses of polynomial endomorphisms.

The Jacobian matrix is nilpotent iff all right powers of the derivation have zero divergence.

If the Jacobian matrix is nilpotent, then the derivation is right nilpotent.

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.