A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

Abstract

We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

On Casimir operators of conformal Galilei algebras

“…In this short paper, we have constructed, via the search algorithm presented in [28], the Casimir operators of certain conformal Galilei algebras with central extensions. The focus was on the cases corresponding to underlying spatial dimensions d = 1 and d = 2, and different values of ℓ (positive half-odd integer or integer).…”

“…The current article has the modest goal of filling this gap for d = 1, 2, and various values of ℓ which allow a central extension. In doing so, we discover certain structural aspects of the conformal Galilei algebras which can improve the functionality of the algorithm presented in [28].…”

“…Following the algorithm, we need to first establish the artificial relative dimensions of the basis elements, a type of grading of the Lie algebra in a certain basis, as discussed in [28]. We introduce artificial relative dimensions α and β and express the dimensions of all generators in a consistent way with the commutation relations (1), (2) and (4).…”

“…The others are quadratic and quartic Casimir operators. The following expression of quadratic Casimir operator of artificial relative dimension (0, 1, 0) was deduced from the forms of the Casimir operators given in (26), (28) and (31), but the proof is a straightforward but tedious exercise in commutation relations.…”

Generalized conformal pseudo-Galilean algebras and their Casimir operators

“…The action of s over the radical is given by the characteristic representation Γ = (D ℓ ⊗ ρ d ) ⊕ Γ 0 , where D ℓ denotes the irreducible representation of sl(2, R) with highest weight 2ℓ and dimension 2ℓ + 1, rho d is the defining d-dimensional representation of so(d) and Γ 0 denotes the trivial representation. Considering the basis (see e.g [17]) given by the generators {H, D, C, E ij = −E ji , P n,i } with n = 0, 1, 2, . .…”