Symmetries in the Hubbard model with n-fold orbital degeneracy

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The present paper studies the symmeries of the Hubbard model of electrons with generally n-fold orbital degeneracy. It's shown SUd(2n) and SUc(2n) symmetries hold respectively for the model with completely repulsive or attractive on-site interaction and that with partly attractive interactions. An extended Lieb-Mattis transformation is given to map these two symmetries into each other. The sub-symmetry SUd⁽ᵉ⁾(n)⊗ SUd⁽ᵒ⁾(n) is found to be shared by the two models with arbitrary chemical potential μ. By assuming at most two electrons on each site it's found that SUd(2n) and SUc(2n) both exist in each kind of the two models and consequently lead to a larger symmetry SUd(2n)× SUc(2n). Another underlying symmetry SUc⁽ᵉ⁾(2) P×...× SUc⁽ᵉ⁾(2) P ⊗ (SUc⁽ᵒ⁾(2) P×...× SUc⁽ᵒ⁾(2) P) is also revealed for the unified U model under the excluding. The symmetry is valid for the partially attractive model with chemical potential μ=-U.