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Mazorchuk's conjecture on mixed modules over the Virasoro algebra

An irreducible weight module V\mathcal{V} is called a mixed module if there exist λ∈C\lambda\in\mathbb{C} and i∈Zi\in\mathbb{Z} such that

dim⁡Vλ=∞anddim⁡Vλ+i<∞.\dim\mathcal{V}_{\lambda}=\infty\quad\text{and}\quad\dim\mathcal{V}_{\lambda+i}<\infty.

Mazorchuk's conjecture. There are no irreducible mixed modules over the Virasoro algebra.

This conjecture concerns the possible weight-space dimensions of irreducible Virasoro-algebra modules: it predicts that an irreducible module cannot have both an infinite-dimensional weight space and another finite-dimensional weight space in the same weight lattice. The source gives no resolution evidence, so its status is left open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Mazorchuk's conjecture on mixed modules over the Virasoro algebra

    An irreducible weight module VV is called a mixed module if there exist [200 λ∈C[200~\lambda\in\mathbb{C} and k∈Z∗k\in\mathbb{Z}^* such that

    dim⁡Vλ=∞anddim⁡Vλ+k<∞.\dim V_\lambda=\infty\quad\text{and}\quad \dim V_{\lambda+k}<\infty.

    Mazorchuk's conjecture. There are no irreducible mixed modules over the Virasoro algebra.

    This conjecture concerns the coexistence of infinite- and finite-dimensional weight spaces in irreducible Virasoro modules. Its resolution is not established by the supplied source context.

    source: Huanxia Fa, Jianzhi Han and Junbo Li, “Irreducible weight modules with a finite-dimensional weight space over the twisted N=1 Schrödinger-Neveu-Schwarz algebra”, arXiv:1703.05072 (2017).

Sources & referencesView supporting material

Primary source

Junbo Li and Yucai Su, “Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over the Twisted Schrödinger-Virasoro Lie algebra”, arXiv:0801.2205 (2008).

representation-theoryvirasoro-algebraweight-modules

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