$\mathcal{W}(a,b)$ Lie Conformal Algebra and Its Conformal Module of Rank One

Abstract

For any complex parameters a, b, let [Formula: see text] be the Lie algebra with basis {Li,Hi | i ∈ ℤ} and relations [Li,Lj]=(j-i)Li+j, [Li,Hj]=(a+j+bi)Hi+j and [Hi,Hj]=0. In this paper, we construct the [Formula: see text] conformal algebra for some a, b and its conformal module of rank one.

A class of super Heisenberg-Virasoro algebras

“…It is well known that all free non-trivial modules of Heisenberg-Virasoro Lie conformal algebra of rank one over C[∂] are as follows (see [33]):…”

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

“…In this paper, we aim to study extensions between conformal modules over a class of Lie conformal algebras W (b), which was introduced in [ where b ∈ C. Finite irreducible conformal modules over W (b) were classified in [8]. It turns out that any Heisenberg-Virasoro conformal algebra in [7]. Its cohomology was studied in [9].…”

Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra

“…In the current paper, we aim to study conformal modules over a family of Lie conformal algebras W(b) introduced in [9] and classify all finite irreducible conformal modules over some Lie conformal algebras related to them. By definition, the Lie conformal algebra W(b) with a parameter b ∈ C is a free Lie conformal algebra generated by L and H as a Moreover, the Lie conformal algebra W(b) has a nontrivial abelian conformal ideal C[∂]H. Thus it is neither simple nor semisimple.…”

“…In [9], the Lie conformal algebra W(b) was called a W(a, b) Lie conformal algebra, because it is closely related to the Lie algebra W (a, b) with a, b ∈ C, which is a semidirect of the centerless Virasoro algebra and a intermediate series module A(a, b). In some special cases, the Lie algebras W (a, b) turn out to be subalgebras of many interesting infinite-dimensional Lie algebras, such as the W-infinity algebra W 1+∞ , the Block and the Schrödinger-Virasoro Lie algebras.…”