Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations

Abstract

ℓ-Conformal Galilei algebra, denoted by g ℓ (d), is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by g ℓ (d) with central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g ℓ (d) and vector field representations for d = 1, 2.

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

“…The operators which solve the on-shell condition induce an invariant PDE for the centrally extended CGA. At degree 1 (or −1) we recover the invariant PDEs obtained in [3]. The novel feature is the invariant operator at degree 0.…”

“…Another possibility, leading to linear invariant PDEs, is based on the construction of singular vectors of the given representation [17] (see [3] for the case of cga ℓ ). We discuss in this Section a different approach to determine linear invariant PDEs, based on imposing an on-shell invariant condition (see [35] and [18]).…”

“…By construction, the two-variable functions ϕ m,n (u, v) which solve the static equation (40) are expressed as products of polynomials in u, v which multiply the ground state function ϕ 0,0 = e 7 The ℓ-oscillator for any ℓ = 1 2 + N 0 Differential realizations of cga ℓ for any half-integer ℓ have been computed in [3]. In that paper the second-order differential operators which, in our language, are on-shell invariant operators of degree 1, were presented.…”

“…The enlarged algebra/superalgebra possesses a differential realization induced by the differential realization of the original CGA (the differential realizations of the latter have been computed, e.g., in [2,3]). The differential realization suitable to the present work requires one time and ℓ + 1 2 space coordinates.…”

“…The PDE (19), derived from the on-shell condition, is identical to the lowest member of the hierarchy, whose construction is based on singular vectors, given in [3]. The on-shell condition allowed us to identify here another invariant operator (Ω 0 , at degree 0), whose importance is discussed in the next Section.…”