Abstract
In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules \(\Omega(\lambda,b)\) with irreducible highest weight modules \(V(\theta ,h)\) or with irreducible Virasoro modules \(\mathrm{Ind}_{\theta}(N)\) defined in Mazorchuk and Zhao (Selecta Math. (N.S.) 20:839–854, 2014). We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of \(\Omega(\lambda,b)\) with a classical Whittaker module is isomorphic to the module \(\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})\) defined in Mazorchuk and Weisner (Proc. Amer. Math. Soc. 142:3695–3703, 2014). As a by-product we obtain the necessary and sufficient conditions for the module \(\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})\) to be irreducible. We also generalize the module \(\mathrm{Ind}_{\theta, \lambda}(\mathbb{C}_{\mathbf{m}})\) to \(\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})\) for any non-negative integer \(n\) and use the above results to completely determine when the modules \(\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})\) are irreducible. The submodules of \(\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})\) are studied and an open problem in Guo et al. (J. Algebra 387:68–86, 2013) is solved. Feigin–Fuchs’ Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.
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Tan, H., Zhao, K. Irreducible Virasoro modules from tensor products. Ark Mat 54, 181–200 (2016). https://doi.org/10.1007/s11512-015-0222-2
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DOI: https://doi.org/10.1007/s11512-015-0222-2