Highest Weight Representations and Kac Determinants for a Class of Conformal Galilei Algebras With Central Extension

Abstract

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrödinger algebra.

Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

“…By Corollary 6.25, (C 1 ,C 2 ) ∈ Prim (U ). Then Prim (U ) ⊇ Prim (U(sl 2 )) ⊔ Max (Z), by (1). Since U/(Z) ≃ U(sl 2 ) and Z(U(sl 2 )) = K[∆], (Z) is not a primitive ideal of U.…”

“…Proof. By Proposition 3.8, the C λ, µ -module V λ, µ (ν 1 ) is not simple if and only if 1 2 (µ − λ 2 ν 1 ) ∈ N + . By Corollary 3.9.…”

“…(2)) and that the algebra U X is a tensor product of three explicit algebras (Proposition 2.4. (1)). This fact is a key in finding the prime spectrum of the algebra U (Theorem 1.1).…”

“…In view of (25), relation (28) can be replaced by the relation 1 and the equalities (12) and (13), the homomorphism in statement 3 is well defined. The fact that it is a monomorphism follows from statement 2 and the claim below.…”

Quasi-Whittaker modules over the conformal Galilei algebras

“…[z,g (l) ] = 0, [p k , p k ] = δ k+k ,2l ( − 1) k+l+ 1 2 k! (2l − k)!z, k, k ∈ 0, 2l, (1.2) Galilei groups and their Lie algebras are important objects in theoretical physics and attract a lot of attention in related mathematical areas.…”