Letters in Mathematical Physics

, Volume 108, Issue 11, pp 2491–2513 | Cite as

Noncommutative products of Euclidean spaces

  • Michel Dubois-Violette
  • Giovanni Landi


We present natural families of coordinate algebras on noncommutative products of Euclidean spaces \({\mathbb {R}}^{N_1} \times _{\mathcal {R}} {\mathbb {R}}^{N_2}\). These coordinate algebras are quadratic ones associated with an \(\mathcal {R}\)-matrix which is involutive and satisfies the Yang–Baxter equations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces \({\mathbb {R}}^{4} \times _{\mathcal {R}} {\mathbb {R}}^{4}\). Among these, particularly well behaved ones have deformation parameter \(\mathbf{u} \in {\mathbb {S}}^2\). Quotients include seven spheres \({\mathbb {S}}^{7}_\mathbf{u}\) as well as noncommutative quaternionic tori \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u} = {\mathbb {S}}^3 \times _\mathbf{u} {\mathbb {S}}^3\). There is invariance for an action of \({{\mathrm{SU}}}(2) \times {{\mathrm{SU}}}(2)\) on the torus \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u}\) in parallel with the action of \(\mathrm{U}(1) \times \mathrm{U}(1)\) on a ‘complex’ noncommutative torus \({\mathbb {T}}^2_\theta \) which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.


Noncommutative Euclidean spaces Noncommutative quaternionic tori Yang–Baxter equations 

Mathematics Subject Classification

16S37 16T25 


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, UMR 8627Université Paris XIOrsay CedexFrance
  2. 2.MatematicaUniversità di TriesteTriesteItaly
  3. 3.INFNTriesteItaly

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