Noncommutative Euclidean spaces

Abstract

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 $T_{\mathbb{H}}^2=U_1(\mathbb{H})\times U_1(\mathbb{H})$

30 pages. arXiv admin note: text overlap with arXiv:1706.06930

References in corpus (2)

• Calabi-Yau algebras
• Nonhomogeneous quadratic duality and curvature

Cited by in corpus (2)

• Principal fibrations over noncommutative spheres
• The $q$-linked complex Minkowski space, its real forms and deformed isometry groups