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**Twisted \(\text{SU}(2)\) group. An example of a non-commutative differential calculus.**
*(English)*
Zbl 0676.46050

Summary: For any number \(\mu\) in the interval \([-1,1]\) a \(C^*\)-algebra \(A\) generated by two elements \(\alpha\) and \(\gamma\) satisfying a simple (depending on \(\mu)\) commutation relation is introduced and investigated.

If \(\mu =1\) then the algebra coincides with the algebra of all continuous functions on the group \(\text{SU}(2)\). Therefore one can introduce many notions related to the fact that \(\text{SU}(2)\) is a Lie group. In particular, one can speak about convolution products, Haar measure, differential structure, cotangent bundle, left invariant differential forms, Lie brackets, infinitesimal shifts and Cartan-Maurer formulae. One can also consider representations of \(\text{SU}(2)\).

For \(\mu <1\) the algebra \(A\) is no longer commutative, however the notions listed above are meaningful. Therefore \(A\) can be considered as the algebra of all “continuous functions” on a “pseudospace \(\text{S}_{\mu}\text{U}(2)\)” and this pseudospace is endowed with a Lie group structure. Potential applications to quantum physics are indicated.

If \(\mu =1\) then the algebra coincides with the algebra of all continuous functions on the group \(\text{SU}(2)\). Therefore one can introduce many notions related to the fact that \(\text{SU}(2)\) is a Lie group. In particular, one can speak about convolution products, Haar measure, differential structure, cotangent bundle, left invariant differential forms, Lie brackets, infinitesimal shifts and Cartan-Maurer formulae. One can also consider representations of \(\text{SU}(2)\).

For \(\mu <1\) the algebra \(A\) is no longer commutative, however the notions listed above are meaningful. Therefore \(A\) can be considered as the algebra of all “continuous functions” on a “pseudospace \(\text{S}_{\mu}\text{U}(2)\)” and this pseudospace is endowed with a Lie group structure. Potential applications to quantum physics are indicated.

### MSC:

46L87 | Noncommutative differential geometry |

46L65 | Quantizations, deformations for selfadjoint operator algebras |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

46L10 | General theory of von Neumann algebras |

### Keywords:

twisted SU(2) group; non-commutative differential calculus; \(C^*\)- algebra; Lie group; convolution products; Haar measure; differential structure; cotangent bundle; left invariant differential forms; Lie brackets; infinitesimal shifts; Cartan Maurer formulae
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\textit{S. L. Woronowicz}, Publ. Res. Inst. Math. Sci. 23, No. 1, 117--181 (1987; Zbl 0676.46050)

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### References:

[1] | Woronowicz, S. L., Compact matrix pseudogroups, in preparation. · Zbl 0627.58034 |

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