Canonical representation of the Snyder-de Sitter algebra with correct flat and commutative limits

The Snyder–de Sitter algebra provides a Lorentz-covariant deformation of phase-space geometry characterized by a curvature parameter $\alpha$ and a noncommutativity parameter $\beta$. We construct an explicit canonical (Darboux) representation of this algebra that is regular in both parameters. Starting from the symplectic structure associated with the Snyder–de Sitter Poisson brackets, we derive the canonical transformation between the physical phase-space variables and Darboux coordinates and obtain its inverse in closed form to all orders in $\alpha$ and $\beta$. The resulting representation has well-defined flat ($\alpha \to 0$) and commutative ($\beta \to 0$) limits, reducing respectively to the Snyder and de Sitter phase-space algebras, while the simultaneous limit $(\alpha ,\beta )\to (0,0)$ yields the standard canonical phase-space coordinates. We then apply this representation to the construction of Poisson gauge transformations on Snyder–de Sitter phase space. In particular, we derive the corresponding gauge transformation matrix and Poisson field strength, both of which are regular in $\alpha$ and $\beta$. Our construction provides a convenient framework for investigating gauge theories and other physical systems formulated on Snyder–de Sitter phase space.