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.


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
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