Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative Euclidean spaces   We give a definition of noncommutative finite-dimensional Euclidean spaces
$\mathbb R^n$. We then remind our definition of noncommutative products of
Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ which produces
noncommutative Euclidean spaces $\mathbb R^{N_1+N_2}$. We solve completely the
conditions defining the noncommutative products of the Euclidean spaces
$\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ and prove that the corresponding
noncommutative unit spheres $S^{N_1+N_2-1}$ are noncommutative spherical
manifolds. We then apply these concepts to define "noncommutative" quaternionic
planes and noncommutative quaternionic tori on which acts the classical
quaternionic torus $T^2_{\mathbb H}=U_1(\mathbb H)\times U_1(\mathbb H)$

30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
16
The categories of this paper are: stat
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph