Twisted Group. An Example of a Non-Commutative Differential Calculus

  • Stanisław Lech Woronowicz

    University of Warsaw, Poland


For any number in the interval , a -algebra , generated by two elements and satisfying simple (depending on ) commutation relation, is introduced and investigated.

If , then the algebra coincides with the algebra of all continuous functions on the group . Therefore, one can introduce many notions related to the fact that is a Lie group. In particular one can speak about convolution products, Haar measure, differential structure, cotangent boundle, left invariant differential forms. Lie brackets, infinitesimal shifts and Cartan Maurer formulae. One can also consider representations of .

For , the algebra is no longer commutative, however the notions listed above are meaningful. Therefore, can be considered as the algebra of all “continuous functions” on a “pseudospace ” and this pseudospace is endowed with a Lie group structure.

The potential applications to the quantum physics are indicated.

Cite this article

Stanisław Lech Woronowicz, Twisted Group. An Example of a Non-Commutative Differential Calculus. Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, pp. 117–181

DOI 10.2977/PRIMS/1195176848