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A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Liealgebra $h(1)\oplus h(1)$, where $h(1)$ is the Heisenberg-Weyl algebra, ispresented in terms of a basis of associated Laguerre polynomials. Using theSchwinger method the quadratic generators that span the alternative Liealgebras $so(3)$, $so(2,1)$ and $so(3,2)$ are also constructed. These familiesof (pseudo) orthogonal algebras also allow to obtain unitary irreduciblerepresentations in $L^2(\mathbb{R}^2)$ similar to those of the sphericalharmonics.

Group theoretical aspects of $L^2(\mathbb{R}^+)$, $L^2(\mathbb{R}^2)$ and associated Laguerre polynomials