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.
Communicated by Volodymyr Mazorchuk MSC: 17B10 17B65 17B68 Keywords: The Virasoro algebra Block type Lie algebras Quasifinite representationsA well-known theorem of Mathieu's states that a Harish-Chandra module over the Virasoro algebra is either a highest weight module, a lowest weight module or a module of the intermediate series. It is proved in this paper that an analogous result also holds for the Lie algebra B related to Block type, with basis {L α,i , C | α, i ∈ Z, i 0} and relations [L α,i , L β, j ] = ((i + 1)β − ( j + 1)α)L α+β,i+ j + δ α+β,0 δ i+ j,0 α 3 −α 12 C . Namely, an irreducible quasifinite B-module is either a highest weight module, a lowest weight module or a module of the intermediate series.
In this paper, a class of Lie conformal algebras associated to a Schrödinger–Virasoro type Lie algebra is constructed, which is nonsimple and can be regarded as an extension of the Virasoro conformal algebra. Then conformal derivations, second cohomology group with trivial coefficients and conformal modules of rank 1 of this Lie conformal algebra are investigated.
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