# Noncommutative products of Euclidean spaces

- 13 Downloads

## Abstract

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.

## Keywords

Noncommutative Euclidean spaces Noncommutative quaternionic tori Yang–Baxter equations## Mathematics Subject Classification

16S37 16T25## References

- 1.Berger, R.: Confluence and quantum Yang-Baxter equation. J. Pure Appl. Algebra
**145**(2000), 267–283 (2000)MathSciNetCrossRefMATHGoogle Scholar - 2.Berger, R.: Dimension de Hochschild des algèbres gradués. C. R. Math. Acad. Sci. Paris
**341**, 597–600 (2005)MathSciNetCrossRefMATHGoogle Scholar - 3.Berger, R., Ginzburg, V.: Higher symplectic reflection algebras and non-homogeneous \({N}\)-Koszul property. J. Algebra
**305**, 577–601 (2006)MathSciNetCrossRefMATHGoogle Scholar - 4.Braverman, A., Gaitsgory, D.: Poincaré–Birkhoff–Witt theorem for quadratic algebras of Koszul type. J. Algebra
**181**, 315–328 (1996)MathSciNetCrossRefMATHGoogle Scholar - 5.Cartan, H.: Homologie et cohomologie d’une algèbre graduée. Séminaire Henri Cartan
**11**(2), 1–20 (1958)Google Scholar - 6.Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys.
**230**, 539–579 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar - 7.Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys.
**221**, 141–159 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar - 8.Dubois-Violette, M.: Multilinear forms and graded algebras. J. Algebra
**317**, 198–225 (2007)MathSciNetCrossRefMATHGoogle Scholar - 9.Dubois-Violette, M.: Poincaré duality for Koszul algebras. Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, vol. 85, pp. 3–26. Springer, Berlin (2014)Google Scholar
- 10.Dubois-Violette, M., Landi, G.: Noncommutative Euclidean spaces. arXiv:1801.03410 [math.QA]
- 11.Ginzburg, V.: Calabi–Yau Algebras. arXiv:math/0612139 [math.AG]
- 12.Gurevich, D.I.: Algebraic aspects of the quantum Yang–Baxter equation. Algebra i Analiz
**2**, 119–148 (1990)MathSciNetMATHGoogle Scholar - 13.Manin, Y.I.: Quantum Groups and Non-commutative Geometry. CRM Université de Montréal, Montreal (1988)MATHGoogle Scholar
- 14.Polishchuk, A., Positselski, L.: Quadratic algebras. University Lecture Series, vol. 37. American Mathematical Society, Providence (2005)MATHGoogle Scholar
- 15.Positselski, L.: Nonhomogeneous quadratic duality and curvature. Funct. Anal. Appl.
**27**, 197–204 (1993)MathSciNetCrossRefGoogle Scholar - 16.Wambst, M.: Complexes de Koszul quantiques. Ann. Inst. Fourier Grenoble
**43**, 1089–1156 (1993)MathSciNetCrossRefMATHGoogle Scholar