A new matrix method for the Casimir operators of the Lie algebras and

Abstract

Abstract.A method is given to determine the Casimir operators of the perfect Lie algebras wsp (N, R) = sp (2N, R) − → ⊕ Γω 1 ⊕Γ0 h N and the inhomogeneous Lie algebras Isp (2N, R) in terms of polynomials associated to a parametrized (2N + 1)× (2N + 1)-matrix. For the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation. The method is extended to other classes of Lie algebras, and some applications to the missing label problem are given.

On Casimir operators of conformal Galilei algebras

“…Various results have already appeared in the literature concerning explicit expressions for Casimir operators for the cases d = 1, 2, 3 and ℓ = 1 2 (e.g. see [20,26,34,35] ). With the help of the orbit method, Casimir operators of conformal Galilei algebras with mass extension have been found in [21] for the case of d = 3 and half-odd integer ℓ.…”

Generalized conformal pseudo-Galilean algebras and their Casimir operators

“…are satisfied, showing that f is a Casimir operator of the radical r. Expanding the condition (12) and taking into account the homogeneity degrees, after a routine computation we find that the system…”

Application of the Gel'fand Matrix Method to the Missing Label Problem in Classical Kinematical Lie Algebras

“…To compute the Casimir operators of g using a matrix extension of A, the following procedure by steps has been proposed [5]:…”

“…As an example of a general class to which the algorithm applies, we consider the inhomogeneous symplectic Lie algebras Isp(2N, R) and the matrix formula obtained for them in [5]. Over the basis {X i,j , X −i,j , X i,j , P i , Q i } (1 ≤ i, j ≤ N ) the brackets of Isp (2N, R) are given by:…”