Smooth Structures, Actions of the Lie Algebra $\mathfrak{su}(2)$ and Haar Measures on Non-Commutative Three Dimensional Spheres
MATSUMOTO, Kengo
Tokyo J. of Math., Tome 15 (1992) no. 2, p. 199-222 / Harvested from Project Euclid
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We study a smooth structure on a non-commutative 3-sphere $S_\Theta^3$ defined as a deformed $C^*$-algebra of $C(S^3)$ by a continuous function $\Theta$. We then consider the subalgebra $(S_\Theta^3)^\infty$ of all smooth elements of $S_\Theta^3$. It is a non-commutative version of $S^{3}$ as a smooth manifold. We also construct a smooth linear map from $(S_\Theta^3)^\infty$ to the algebra $C^\infty(S^3)$ of all smooth functions on $S^3$ so that the Lie algebra $\mathfrak{su}(2)$ acts on $(S_\Theta^3)^\infty$ with a twisted Leibniz's rule. Finally we find a Haar measure on $S_\Theta^3$ and show its uniqueness.
Publié le : 1992-06-15
Classification:  
@article{1270130261,
     author = {MATSUMOTO, Kengo},
     title = {Smooth Structures, Actions of the Lie Algebra $\mathfrak{su}(2)$ and Haar Measures on Non-Commutative Three Dimensional Spheres},
     journal = {Tokyo J. of Math.},
     volume = {15},
     number = {2},
     year = {1992},
     pages = { 199-222},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1270130261}
}

MATSUMOTO, Kengo. Smooth Structures, Actions of the Lie Algebra $\mathfrak{su}(2)$ and Haar Measures on Non-Commutative Three Dimensional Spheres. Tokyo J. of Math., Tome 15 (1992) no. 2, pp.  199-222. http://gdmltest.u-ga.fr/item/1270130261/