Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Associative, Lie, and left-symmetric algebras of derivations   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.

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The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
8
The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn