[Paper Review] A class of simple weight Virasoro modules

Genqiang Liu, Rencai Lü+1 more|arXiv (Cornell University)|Nov 5, 2012

Algebraic structures and combinatorial models17 references3 citations

TL;DR

This paper constructs a new class of simple weight Virasoro modules, denoted $\mathcal{N}(M,\alpha)$, by inducing from irreducible modules $M$ over finite-dimensional solvable Lie algebras $\mathfrak{a}_r$, with $r \geq 1$. The key result is a necessary and sufficient condition for simplicity and isomorphism, yielding infinitely many new irreducible weight and nonweight Virasoro modules, including explicit examples on $\mathbb{C}[x] \otimes \mathbb{C}[t,t^{-1}]$ and nonweight modules via twisting.

ABSTRACT

For a simple module $M$ over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras $\mathfrak{a}_r$) and any $α\in\C$, a class of weight modules $\mathcal {N}(M, α)$ over the Virasoro algebra are constructed. The necessary and sufficient condition for $\mathcal {N}(M, \a)$ to be simple is obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules $Γ(α_1, α_2, λ_1, λ_2)$ are defined on the polynomial algebra $\C[x]\otimes \C[t, t^{-1}]$ for any $α_1, α_2, λ_1, λ_2\in\C$ with $λ_1$ or $λ_2$ nonzero. By twisting the weight modules $\mathcal {N}(M, α)$ we also obtain nonweight simple Virasoro modules $\mathcal {N}(M, β)$ for any $β\in\C[t,t^{-1}]$.

Motivation & Objective

• To construct a large class of new irreducible weight Virasoro modules with infinite-dimensional weight spaces.
• To provide necessary and sufficient conditions for such modules to be simple and to classify their isomorphism classes.
• To extend the construction to nonweight irreducible Virasoro modules via twisting.
• To demonstrate that the new modules are not isomorphic to any previously known irreducible Virasoro modules.
• To provide explicit realizations of such modules on polynomial algebras, including $\Gamma(\alpha_1,\alpha_2,\lambda_1,\lambda_2)$.

Proposed method

• Define a Virasoro module structure on $\mathcal{N}(M,\alpha) = M \otimes \mathbb{C}[t^{\pm 1}]$ using the action $d_m(v \otimes t^n) = \left(\alpha + n + \sum_{i=0}^r \frac{m^{i+1}}{(i+1)!} \bar{d}_i\right)v \otimes t^{n+m}$, with $c$ acting as zero.
• Prove that $\mathcal{N}(M,\alpha)$ is a Virasoro module by verifying the Lie bracket relations, particularly the commutator $[d_m, d_k]$.
• Establish that $\mathcal{N}(M,\alpha)$ is simple if $M$ is an infinite-dimensional irreducible $\mathfrak{a}_r$-module and $r \geq 1$, using the irreducibility of $M$ and the action of $d_i$ operators.
• Use the operator $\omega_{l,m}^{(2r+2)}$ to distinguish the new modules from known ones, showing nonvanishing action on nonzero vectors.
• Construct nonweight modules $\mathcal{N}(M,\beta)$ by twisting the weight modules with $\beta \in \mathbb{C}[t,t^{-1}]$, preserving simplicity.
• Compare the new modules with known families: intermediate series, tensor products, Whittaker modules, and Weyl modules, proving non-isomorphism via distinct operator actions.

Experimental results

Research questions

• RQ1 Under what conditions is the constructed module $\mathcal{N}(M,\alpha)$ simple over the Virasoro algebra?
• RQ2 When are two such modules $\mathcal{N}(M,\alpha)$ and $\mathcal{N}(M',\alpha')$ isomorphic as Virasoro modules?
• RQ3 Are the new weight modules $\mathcal{N}(M,\alpha)$ isomorphic to any previously known irreducible Virasoro modules with infinite-dimensional weight spaces?
• RQ4 Can the construction be extended to produce nonweight irreducible Virasoro modules, and are they new?
• RQ5 Do the new modules $\mathcal{N}(M,\beta)$ satisfy distinct operator identities compared to known nonweight modules like those in [LZ], [MW], or [MZ3]?

Key findings

• The module $\mathcal{N}(M,\alpha)$ is simple if and only if $M$ is an infinite-dimensional irreducible module over $\mathfrak{a}_r$ and $r \geq 1$, with $\alpha \in \mathbb{C}$ arbitrary.
• Two such irreducible modules $\mathcal{N}(M,\alpha)$ and $\mathcal{N}(M',\alpha')$ are isomorphic if and only if $M \cong M'$ as $\mathfrak{a}_r$-modules and $\alpha = \alpha'$.
• The irreducible weight Virasoro module $\Gamma(\alpha_1,\alpha_2,\lambda_1,\lambda_2)$ is realized on $\mathbb{C}[x] \otimes \mathbb{C}[t,t^{-1}]$ for $\lambda_1$ or $\lambda_2$ nonzero.
• Nonweight irreducible Virasoro modules $\mathcal{N}(M,\beta)$ are constructed for any $\beta \in \mathbb{C}[t,t^{-1}]$, and are shown to be new when $\dim M > 1$ and $\beta \notin \mathbb{C}$.
• The new modules $\mathcal{N}(M,\beta)$ are not isomorphic to any known nonweight modules from [LZ], [MW], or [MZ3], as they satisfy distinct identities under the action of $\omega_{l,m}^{(2r+2)}$ operators.
• The action of $\omega_{l,m}^{(2r+2)}$ on $\mathcal{N}(M,\beta)$ is nonzero for all nonzero vectors, while it vanishes on known modules like those in [MZ3] or [LZ], proving non-isomorphism.