The interaction of various algebraic structures describing fusion, braiding, and group symmetries in quantum projective field theory is the object of investigation in this article. Structures of projective Zamolodchikov algebras, their representations, spherical correlation functions, correlation characters and enveloping quantum projective field theory (QPFT)-operator algebras, projective Ẅ-algebras, shift algebras, infinite-dimensional R matrices Rproj(u) and Rproj*(u) of the QPFT, braiding admissible QPFT-operator algebras, and projective G hypermultiplets are explored. It is proven (in the formalism of shift algebras) that sl(2,C)-primary fields are characterized by their projective weights and by the hidden additive weight, a hidden quantum number discovered in this article. Special attention is paid to various constructions of projective G hypermultiplets (QPFT-operator algebras with G symmetries).