MathDB

Mazorchuk's conjecture on mixed modules over the Virasoro algebra

About 18 years old · traced to

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 1Other wordings

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).

References

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
Posted

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.