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Noncommutative products of Euclidean spaces
Quantum Algebra · Submitted 21 Jun 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We present natural families of coordinate algebras of noncommutative products of Euclidean spaces.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. | contribution 1.00 | 0.94 | 0.38 | asserted 1.00 |
| s1 | These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. | contribution 0.73 | 0.66 | 0.14 | asserted 1.00 |
| s2 | As a consequence they enjoy a list of nice properties, being regular of finite global dimension. | result 0.79 | 0.74 | 0.18 | asserted 1.00 |
| s3 | Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are deformations parametrised by a two-dimensional sphere. | result 0.86 | 0.92 | 0.21 | asserted 1.00 |
| s4 | Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". | result 0.88 | 0.82 | 0.09 | asserted 1.00 |
| s5 | 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. | result 0.76 | 0.79 | 0.10 | asserted 1.00 |
| s6 | Additional classes of solutions are disjoint from the classical case. | result 0.92 | 0.79 | 0.09 | asserted 1.00 |
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Original 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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How the literature uses it
3 arXiv papers cite this in 7 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 3 papers
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In Section [ref] we recall from [ref] and [ref] results on the quadratic algebras that we need and some of the solutions for noncommutative spheres $^7_R$ that we use later on (here $R$ is a matrix of deformation parameters).
Principal fibrations over noncommutative spheres · Introduction and motivations · 0.88
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Their representations have been studied in [ref] .
Fock representations of Zamolodchikov algebras and R-matrices · An Abstract Zamolodchikov-Faddeev Algebra · 0.90
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Such involutive R-matrices play a prominent role in various fields, ranging from symmetries of categories of vector spaces [ref] over scattering operators in integrable quantum field theory (with a spectral parameter) [ref] to representations of Thompson's group $V$ [ref] and recent constructions of non-commutative spaces [ref] .
Yang-Baxter representations of the infinite symmetric group · Introduction · 0.88
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