Classification of finite irreducible conformal modules over a class of Lie conformal algebras of Block type

Abstract

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over B(p) may be a nontrivial extension of a finite conformal module over Vir if p = −1,where Vir is a Virasoro conformal subalgebra of B(p). As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras b(n) for n ≥ 1.

Structure of a class of Lie conformal algebras of Block type

“…In particular, the case B(1) is a maximal subalgebra of the associated graded conformal algebra grgc 1 of the filtered algebra gc 1 [14]. Besides, for any integer n ≥ 1, B(−n) contains a series of finite Lie conformal quotient algebras (see [13], Section 2.2).…”

“…The Lie algebra A(B(p)) is related to the Lie algebra B(q) of Block type [11,12], Hence, this Lie conformal algebra B(p) is called a Lie conformal algebra of Block type [13].…”

“…One can see that the one-dimensional vector space C can be regarded as a module (called a trivial module) over B(p) with the action of ∂,L i being zero. Recall from [13] that there are some ∆, α, β ∈ C such that (1) If p = −1, then M ∼ = M ∆,α , where…”

“…More recently, in [13], the authors introduced another class of infinite rank Lie conformal algebras B(p), whose annihilation algebras have close relation with the Lie algebras B(q) of * Supported by a grant from China Scholarship Council (201708645021), National Natural Science Foundation grants of China (11661063,11401570), and the Fundamental Research Funds for the Central Universities (2019QNA34).…”

Extensions of Schrödinger–Virasoro conformal modules

“…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.…”