[Paper Review] New irreducible tensor product modules for the Virasoro algebra

Xiangqian Guo, Xuewen Liu+1 more|arXiv (Cornell University)|Aug 30, 2017

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper constructs new irreducible non-weight Virasoro modules by taking tensor products of irreducible modules $Ω(\lambda,\alpha,h)$ with irreducible highest weight modules $V(\theta,h)$ or modules $\mathrm{Ind}_\theta(N)$, establishing necessary and sufficient conditions for irreducibility and isomorphism. The key contribution is proving these modules are new and not isomorphic to any previously known irreducible Virasoro modules.

ABSTRACT

In this paper, we obtain a class of Virasoro modules by taking tensor products of the irreducible Virasoro modules $Ω(λ,α,h)$ defined in \cite{CG}, with irreducible highest weight modules $V(θ,h)$ or with irreducible Virasoro modules Ind$_θ(N)$ defined in \cite{MZ2}. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for two of them to be isomorphic. These modules are not isomorphic to any other known irreducible Virasoro modules.

Motivation & Objective

• To construct new classes of irreducible non-weight Virasoro modules through tensor products of known irreducible modules.
• To determine necessary and sufficient conditions for such tensor product modules to be irreducible.
• To establish conditions under which two such tensor product modules are isomorphic.
• To prove that the constructed modules are not isomorphic to any other known irreducible Virasoro modules, thus identifying them as new.
• To reformulate the tensor modules as induced modules from subalgebras of the Virasoro algebra for structural insight.

Proposed method

• Define the Virasoro modules $\Omega(\lambda,\alpha,h)$ as free $\mathbb{C}[t,s]$-modules with specific actions of $d_m$ on $f(t)s^i$.
• Tensor $\Omega(\lambda,\alpha,h)$ with irreducible highest weight modules $V(\theta,h)$ or with $\mathrm{Ind}_\theta(N)$, which are locally finite over the positive part of the Virasoro algebra.
• Use the PBW theorem to construct bases for the induced modules $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C}[t]_{\alpha,h,\mathbf{a}})$ and compare them with the tensor product module bases.
• Define a linear map $\phi$ from the induced module to the tensor product module and prove it is a Virasoro module isomorphism via verifying module homomorphism and bijectivity.
• Leverage known irreducibility criteria for $\Omega(\lambda,\alpha,h)$ (when $\deg(h)=1$ and $\alpha \neq 0$) and for $V_{\mathbf{a},\theta}$ (when $a_{2n-1}^2 + a_{2n}^2 \neq 0$) to determine irreducibility of the tensor product.
• Use total order on basis elements to prove injectivity of the isomorphism $\phi$, ensuring the tensor product module is irreducible if and only if both factors are irreducible.

Experimental results

Research questions

• RQ1 Under what conditions is the tensor product $\Omega(\lambda,\alpha,h) \otimes V(\theta,h)$ irreducible?
• RQ2 When is the tensor product $\Omega(\lambda,\alpha,h) \otimes \mathrm{Ind}_\theta(N)$ irreducible?
• RQ3 What are the necessary and sufficient conditions for two such tensor product modules to be isomorphic?
• RQ4 Are these new tensor product modules isomorphic to any previously known irreducible Virasoro modules?
• RQ5 Can the tensor product modules be realized as induced modules from subalgebras of the Virasoro algebra?

Key findings

• The tensor product $\Omega(\lambda,\alpha,h) \otimes V(\theta,h)$ is irreducible if and only if $\deg(h) = 1$ and $\alpha \neq 0$, and $V(\theta,h)$ is irreducible.
• The tensor product $\Omega(\lambda,\alpha,h) \otimes \mathrm{Ind}_\theta(N)$ is irreducible if and only if $\deg(h) = 1$, $\alpha \neq 0$, and $a_{2n-1}^2 + a_{2n}^2 \neq 0$.
• Two irreducible tensor product modules $\Omega(\lambda,\alpha,h) \otimes V(\theta,h)$ are isomorphic if and only if their parameters $\lambda, \alpha, h, \theta$ satisfy specific equivalence conditions derived from the module structures.
• The constructed modules are not isomorphic to any other known irreducible Virasoro modules, establishing their novelty.
• The tensor product module $\Omega(\lambda,\alpha,h) \otimes V(\mathbf{a},\theta)$ is isomorphic to the induced module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C}[t]_{\alpha,h,\mathbf{a}})$, providing a new realization of these modules.
• The isomorphism $\phi$ between $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C}[t]_{\alpha,h,\mathbf{a}})$ and $\Omega(\lambda,\alpha,h) \otimes V(\mathbf{a},\theta)$ is established via a basis-preserving, Virasoro-equivariant map with injective and surjective properties proven using total order on monomials.