XXXII Workshop on Geometric Methods in Physics | 30.06-6.07.2013 |

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## Marat Aukhadiev## Classical compact quantum semigroups as deformations of abelian groupsThe algebra of continuous functions $C(G)$ of a compact abelian group $G$ can be deformed using its Pontryagin dual discrete group $\Gamma$. In fact, this new C*-algebra is generated by an inverse semigroup, and is called a reduced semigroup C*-algebra. We show that this C*-algebra can be regarded as an algebra of functions on a compact quantum semigroup $QS$, which posesses different interesting properties. Thus, $QS$ is a "deformation" of the group "G". The quantum semigroup $QS$ is endowed with a natural coaction of $G$, given by a C*-dynamical system. Such compact quantum semigroups form a tensor category, dual to a category of some abelian semigroups. |

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