Extensions of modules over a class of Lie conformal algebras 𝒲(b)

Abstract

Let W (b) be a class of free Lie conformal algebras of rank 2 with C[∂ ]-basis {L, H} and relationswhere b is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras W (b).

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

“…Applying both sides of (3.1) and (3.2) to v α gives the following equations: Then we obtain the result by Theorem 2.7. (3.26) By Lemma 3.6 in [13], we obtain all solutions of (3.26) as follows.…”

“…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).…”

“…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] The rest of this paper is organized as follows. In Section 2, we introduce some basic definitions, notations, and related known results about Virasoro Lie conformal algebra V ir.…”

“…(Ref [13],. Theorem 3.7) Nontrivial extensions of finite irreducible W(a, 0)modules of the form(3.19) with a = 1 exist if and only if β =β.…”

Classification of finite irreducible conformal modules over Lie conformal algebra <inline-formula><tex-math id="M1">$ \mathcal{W}(a, b, r) $</tex-math></inline-formula>

“…From a semi-direct sum of the centerless Virasoro algebra and its intermediate series module A(a, b), a class of Lie conformal algebras W(b) of rank two was first obtained in [17], and their conformal modules of rank one were also constructed. Finite irreducible conformal modules over W(b) were classified in [16] and of the extensions in [12]. A more general class of Lie conformal algebras W(a, b), constructed as a semi-direct sum of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one, was introduced in [14].…”