Three algebraic structures of quantum projective ($\mathrm{sl}(2,\mathbb C)$-invariant) field theory

Systematic studies are made of three algebraic structures of quantum projective ($\mathrm{sl}(2,\mathbb C)$-invariant) field theory: the operator algebra $\mathrm{Vert}(\mathrm{sl}(2,\mathbb C))$, the infinite dimensional $R$-matrix $R_{\mathrm{proj}}(u)$ and deformation $\mathcal T_\hbar(\mathbb R)$ of the algebra $\mathcal T(\mathbb R)$ of weighted-shift operators, which is associated with expansion of the renormalized pointwise product of vertex operator fields.