We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for all multipartite symmetric states in the general N-qubit case. For this purpose, we introduce 2 parameters playing a crucial role, namely, the diversity degree and the degeneracy configuration of a symmetric state. Those parameters give rise to a simple method of identifying operational families of SLOCC entanglement classes of all symmetric N-qubit states, where the number of families grows as the partition function of the number of qubits.

We study the interconversion of multipartite symmetric N -qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric N -qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states.

The existence and uniqueness of quantizations that are equivariant with respect to conformal and projective Lie algebras of vector fields were recently obtained by Duval, Lecomte and Ovsienko. In order to do so, they computed spectra of some Casimir operators. We give an explicit formula for those spectra in the general framework of IFFTalgebras classified by Kobayashi and Nagano. We also define tree-like subsets of eigenspaces of those operators in which eigenvalues can be compared to show the existence of IFFT-equivariant quantizations. We apply our results to prove existence and uniqueness of quantizations that are equivariant with respect to the infinitesimal action of the symplectic (resp. pseudo-orhogonal) group on the corresponding Grassmann manifold of maximal isotropic subspaces.Math. Classification (AMS 2000): 17B66, 22E46, 81R05. Keywords: Lie subalgebras of vector fields, Modules of differential operators, Casimir operators.

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