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Noncommutative geometry of angular momentum space U(su(2))
High Energy Physics - Theory · Submitted 14 May 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We embed $R^3_\lambda$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_\lambda$ has a natural quantum isometry group given by the quantum double $D(U(su(2)))$ as a singular limit of the $q$-Lorentz group.
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The abstract, sentence by sentence
We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator. We embed $R^3_\lambda$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_\lambda$ has a natural quantum isometry group given by the quantum double $D(U(su(2)))$ as a singular limit of the $q$-Lorentz group. We view $\R^3_\lambda$ as a collection of all fuzzy spheres taken together. We also analyse the semiclassical limit via minimum uncertainty states $|j,\theta,\phi>$ approximating classical positions in polar coordinates.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. | contribution 0.82 | 0.79 | 0.09 | asserted 1.00 |
| s1 | We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. | contribution 0.63 | 0.96 | 0.13 | asserted 1.00 |
| s2 | We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator. | result 0.56 | 0.96 | 0.13 | asserted 1.00 |
| s3 | We embed $R^3_\lambda$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_\lambda$ has a natural quantum isometry group given by the quantum double $D(U(su(2)))$ as a singular limit of the $q$-Lorentz group. | contribution 0.57 | 0.94 | 0.32 | asserted 1.00 |
| s4 | We view $\R^3_\lambda$ as a collection of all fuzzy spheres taken together. | interpretation 0.94 | 0.77 | 0.11 | asserted 1.00 |
| s5 | We also analyse the semiclassical limit via minimum uncertainty states $|j,\theta,\phi>$ approximating classical positions in polar coordinates. | contribution 0.83 | 0.93 | 0.07 | asserted 1.00 |
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The paper
Original abstract
We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator. We embed $R^3_\lambda$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_\lambda$ has a natural quantum isometry group given by the quantum double $D(U(su(2)))$ as a singular limit of the $q$-Lorentz group. We view $\R^3_\lambda$ as a collection of all fuzzy spheres taken together. We also analyse the semiclassical limit via minimum uncertainty states $|j,\theta,\phi>$ approximating classical positions in polar coordinates.
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References
How the literature uses it
10 arXiv papers cite this in 18 sentences
Papers citing it · 12 sentences without a confident label
Cites it as background 6 papers
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This space is usually denoted $^3_λ$ in the literature [ref] .
Quantized Nambu-Poisson Manifolds and n-Lie Algebras · $_0^3$ · 0.87
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A $2d$-dimensional Lie algebra $𝔞$ has the structure of a Drinfel'd double (DD) [ref] (see also [ref] ) if there exists a basis ${Y_1,…,Y_d,y^1,…,y^d }$ of $𝔞$ in which the Lie bracket takes the form [equation] Note that this implies that ${Y_1,…,Y_d}$ and ${y^1,…,y^d }$ span two Lie subalgebras with structure constants $c^k_ij$ and $f^ij_k$, respectively.
Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes · Drinfel'd double Lie algebras · 0.99
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Examples of NC spaces where $Θ_μν$ [ref] do not depend on the momenta $p_μ$ are the canonical theta space [ref] , with $Θ_μν = const$, Lie algebra type spaces [ref] , for which $Θ_μν = i C_μνλ x̂_λ$, and quadratic deformations of Minkowski space, with $Θ_μν = Θ^μνρσ x̂_ρx̂_σ$ [ref] .
Deformed Quantum Phase Spaces, Realizations, Star Products and Twists · Introduction · 0.95
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