= Solution
In $D$ dimensions the antisymmetric <angular momentum tensor> has components
$$
L_{ab}=\sum_{s=1}^2(q_{s,a}p_{s,b}-q_{s,b}p_{s,a}),\qquad a<b.
$$
There are $D(D-1)/2$ independent rotation generators. Together with $D$ translation generators and one time-translation generator, the standard <Noether charge> enumeration is therefore
$$
\boxed{1+D+\frac{D(D-1)}2=1+\frac{D(D+1)}2.}
$$
The word independent here counts symmetry generators and tensor components; it needs qualification if it is meant as functional independence of all charges for only two particles in arbitrarily high dimension. In <centre-of-mass coordinates>, $L=R\wedge P+r\wedge p$. Consequently $P\wedge L\wedge L=0$, a nontrivial identity from dimension five onwards, since $(r\wedge p)\wedge(r\wedge p)=0$. Thus the displayed number is the symmetry-charge count; it is not an assertion that these functions have unrestricted values for a two-particle realization in every $D$.
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