Extensions of Schrödinger–Virasoro conformal modules

Abstract

In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger-Virasoro conformal algebra and the extended Schrödinger-Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger-Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg-Virasoro conformal modules.

Finite irreducible modules of Lie conformal algebras 𝒲(a,b) and some Schrödinger–Virasoro type Lie conformal algebras

“…Actually, W(1, 0) is the Heisenberg-Virasoro conformal algebra. Moreover, extensions of finite irreducible modules over it were classified in [12,19]. So we will give their results directly below without proof.…”

“…On the other hand, it follows from (3.15) that [19]. (3.14) exist if and only if β + η = 0 and α = 1.…”

“…(2) See Corollary 6.3 in [19]. Now, we can consider nontrivial extensions of finite irreducible W(a, b)-modules of the form (3.19) when (a, b) = (1, 0).…”

“…Assume that g(∂, λ) = a 31 ∂ 2 λ + a 32 ∂λ 2 + a 33 λ 3 + a 20 ∂ 2 + a 21 ∂λ + a 22 λ 2 + a 10 ∂ + a 11 λ + a 00 . Plugging this into (3.27) and using undetermined coefficient method, we can obtain that g(∂, λ) = ∂ 2 λ + 3∂λ 2 + 2λ 3 Then nontrivial extensions of finite irreducible W(1, 0)-modules of the form (3.19) were classified by Yuan and Ling in Corollary 6.3 in [19].…”

“…In [11], we gave a complete classification of finite irreducible conformal modules of W(a, b), T SV (a, b) and T SV (c). In [12,13,19], Ling and Yuan classified all extensions of finite irreducible conformal modules over W(1, 0) , W(1 − b, 0) and T SV ( 3 2 , 0). In this paper, we deal with the same problem for W(a, b), T SV (a, b) and T SV (c).…”