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The $N$-dimensional Smorodinsky-Winternitz system is a maximally superintegrable and exactly solvable model, being subject of study from different approaches. The model has been demonstrated to be multiseparable with wavefunctions given by Laguerre and Jacobi polynomials. In this paper we present the complete symmetry algebra ${cal SW}(N)$ of the system, which it is a higher-rank quadratic one containing the recently discovered Racah algebra ${cal R}(N)$ as subalgebra. The substructures of distinct quadratic ${cal Q}(3)$ algebras and their related Casimirs are also studied. In this way, from the constraints on the oscillator realizations of these substructures, the energy spectrum of the $N$-dimensional Smorodinsky-Winternitz system is obtained. We show that ${cal SW}(N)$ allows different set of substructures based on the Racah algebra ${cal R}({ N})$ which can be applied independently to algebraically derive the spectrum of the system.
n-ary algebras have played important roles in mathematics and mathematical physics. The purpose of this paper is to construct a deformation of Virasoro-Witt n-algebra based on an oscillator realization with two independent parameters (p, q) and investigate its n-Lie subalgebra.
We construct the general solution of a class of Fuchsian systems of rank $N$ as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is give
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The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons
We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and