Abstract. We introduce a class of compact quantum semigroups, that we call semigroup deformations of compact Abelian qroups. These objects arise from reduced semigroup C * -algebras, the generalization of the Toeplitz algebra. We study quantum subgroups, quantum projective spaces and quantum quotient groups for such objects, and show that the group is contained as a compact quantum subgroup in the deformation of itself. The connection with the weak Hopf algebra notion is described. We give a grading on the C * -algebra of the compact quantum semigroups constructed.
PreliminariesThis work is devoted to constructing compact quantum semigroups using non-commutative C * -algebras, generated by isometries. The procedure starts with considering a compact Abelian group G and its Pontryagin dual group Γ. The method described in section 2 represents a sort of deformation of the group G, which preserves its topology and multiplication.Unlike the standard quantization procedure, the deformation parameter here is not a number, but a set of elements. More precisely, any semigroup S generating group Γ can play a role of deformation parameter. It is shown that this deformation depends essentially on the choice of S, giving different results for different semigroups generating the same group G.Another special feature of this method is that the object QS r (S) obtained as deformation is not a quantum group, but a compact quantum semigroup with some unique properties. Notion of the compact quantum semigroup is given in section 1.1. We show that QS r (S) can be regarded as an extension of G, since G ⊂ QS r (S) as a compact quantum subgroup (see section 3). Similarly to the classical case this extension generates an action of G on QS r (S). This action and quantum projective space are described in section 3.