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arXiv · 1706.06930

Noncommutative products of Euclidean spaces

Michel Dubois-Violette·Giovanni Landi
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Abstract

We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an 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 which are particularly well behaved and are deformations parametrised by a two-dimensional sphere. Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". There is invariance for an action of $SU(2) \times SU(2)$ in parallel with the action of $U(1) \times U(1)$ on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.

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BibTeXRIS

Michel Dubois-Violette, Giovanni Landi. 2017-11-12. Noncommutative products of Euclidean spaces. https://doi.org/10.1007/s11005-018-1090-z

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