Twisted \(\text{SU}(2)\) group. An example of a non-commutative differential calculus
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.
scientific article; zbMATH DE number 4107479