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

Abstract

In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras W(b) with b ∈ C related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over W(b) is proved to be isomorphic to M ∆,α,β with ∆ = 0 or β = 0 if b = 0, or M ∆,α with ∆ = 0 if b = 0. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the Lie conformal algebra of W(2, 2)-type are classified. Finally, the same thing is done for the Schrödinger-Virasoro conformal algebra.

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

“…The following result, due to [8], gives all finite free irreducible conformal modules over the 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].…”

Extensions of Schrödinger–Virasoro conformal modules

“…where the values of ∆ and∆ along with the corresponding polynomials f (∂ , λ ) whose nonzero scalar multiples give rise to nontrivial extensions are listed as follows (∂ = ∂ + α): All finite nontrivial irreducible conformal modules over the Schrödinger-Virasoro conformal algebra were classified in [13], and the corresponding results are the following. Proposition 2.10.…”

“…In particular, λ -brackets arise as generating functions for the singular part of the OPE. The structure, cohomology and representation theory of LCAs was developed by V. Kac and his coworkers in the late 1990s ( [1][2][3][4][5][6]), and non-semisimple LCAs associated to infinitedimensional Lie algebras of Virasoro type were studied recently in [10][11][12][13][14][15]. As pointed out in [2], conformal modules of LCAs are not completely reducible in general.…”

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

“…Note that W(1 − b, 0) is just the Lie conformal algebra W(b) in [18], W(1, 0) is just the Heisenberg-Virasoro Lie conformal algebra, T SV ( 3 2 , 0) is just the Schrödinger-Virasoro Lie conformal algebra in [15] and T SV (0, 0) is just the Schrödinger-Virasoro type Lie conformal algebra in [16]. Finite irreducible conformal modules of W(1, 0) and W(1 − b, 0) were classified in [17]. In [11], we gave a complete classification of finite irreducible conformal modules of W(a, b), T SV (a, b) and T SV (c).…”