Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Zero product determined Lie algebras   A Lie algebra $L$ over a field $\mathbb{F}$ is said to be zero product
determined (zpd) if every bilinear map $f:L\times L\to \mathbb{F}$ with the
property that $f(x,y)=0$ whenever $x$ and $y$ commute is a coboundary. The main
goal of the paper is to determine whether or not some important Lie algebras
are zpd. We show that the Galilei Lie algebra $\mathfrak{sl}_2\ltimes V$, where
$V$ is a simple $\mathfrak{sl}_2$-module, is zpd if and only if $\dim V =2$ or
$\dim V$ is odd. The class of zpd Lie algebras also includes the quantum torus
Lie algebras $\mathcal{L}_q$ and $\mathcal{L}^+_q$, the untwisted affine Lie
algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at
most $3$, while the class of non-zpd Lie algebras includes the
($4$-dimensional) aging Lie algebra $\mathfrak {age}(1)$ and all Lie algebras
of dimension more than $3$ in which only linearly dependent elements commute.
We also give some evidence of the usefulness of the concept of a zpd Lie
algebra by using it in the study of commutativity preserving linear maps.

37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin