[Paper Review] A family of simple weight modules over the Virasoro algebra

Rencai Lü, Kaiming Zhao|arXiv (Cornell University)|Mar 4, 2013

Algebraic structures and combinatorial models32 references16 citations

TL;DR

This paper constructs a new family of simple weight modules over the Virasoro algebra by inducing from simple modules over the derivation algebra $χ[t]\frac{d}{dt}$, generalizing earlier constructions. It provides necessary and sufficient conditions for simplicity and isomorphism, and proves these modules are new and not isomorphic to any submodules of tensor products of highest weight and intermediate series modules.

ABSTRACT

Using simple modules over the derivation Lie algebra $C[t]\frac{d}{d t}$ of the associative polynomial algebra $C[t]$, we construct new weight Virasoro modules with all weight spaces infinite dimensional. We determine necessary and sufficient conditions for these new weight Virasoro modules to be simple, and determine necessary and sufficient conditions for two such weight Virasoro modules to be isomorphic. If such a weight Virasoro module is not simple, we obtain all its submodules. In particular, we completely determine the simplicity and the isomorphism classes of the weight modules defined in [C. Conley, C. Martin; A family of irreducible representations of the Witt Lie algebra with infinite-dimensional weight spaces. Compos. Math., 128(2), 153-175(2001)] which are a small portion of the modules constructed in this paper.

Motivation & Objective

• To construct a new family of simple weight modules over the Virasoro algebra with all weight spaces infinite-dimensional.
• To generalize and unify previous constructions, including those in [CM] and [LLZ], as special cases.
• To determine necessary and sufficient conditions for such modules to be simple and for two such modules to be isomorphic.
• To prove that the constructed modules are not isomorphic to any simple submodules of tensor products of highest weight and intermediate series modules.
• To classify all submodules when the modules are not simple, providing a complete structural description.

Proposed method

• Construct the Virasoro modules $\mathcal{L}(W,\lambda,a,b) = W \otimes \mathbb{C}[t,t^{-1}]$ where $W$ is a simple module in $\mathcal{O}_{\mathfrak{W}}$, the category of $\mathfrak{W}$-modules satisfying a finiteness condition.
• Use the structure of the Witt algebra $\mathfrak{W} = \operatorname{Der}(\mathbb{C}[t])$ and its subalgebras $\mathfrak{b} = \operatorname{span}\{d_i \mid i \geq 0\}$ and $\mathfrak{V}^{(r)}$ to define the modules via induction.
• Apply a powerful technical tool: defining homomorphisms $\varphi_n$ between weight spaces to analyze module structure and isomorphism types.
• Utilize the operator $X_{l,m} = d_{l-m-3}d_{m+3} - 3d_{l-m-2}d_{m+2} + 3d_{l-m-1}d_{m+1} - d_{l-m}d_m$ to distinguish the new modules from known ones.
• Analyze the action of $X_{l,m}$ on weight vectors to show non-vanishing for certain $\lambda \neq 1$, proving non-isomorphism to tensor product modules.
• Use the socle and order of vectors in $\mathfrak{b}$-modules to classify simple modules in $\mathcal{O}_{\mathfrak{b}}$ and $\mathcal{O}_{\mathfrak{W}}$.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for the constructed weight modules $\mathcal{L}(W,\lambda,a,b)$ to be simple?
• RQ2 When are two such modules $\mathcal{L}(W,\lambda,a,b)$ and $\mathcal{L}(W',\lambda',a',b')$ isomorphic?
• RQ3 How do the submodules of non-simple $\mathcal{L}(W,\lambda,a,b)$ modules look, and can they be completely classified?
• RQ4 Are the constructed modules new, i.e., not isomorphic to any simple submodules of tensor products of highest weight and intermediate series modules?
• RQ5 How do the modules in [CM] and [LLZ] relate to the general family $\mathcal{L}(W,\lambda,a,b)$?

Key findings

• The modules $\mathcal{L}(W,\lambda,a,b)$ are simple if and only if $\lambda \neq 1$ and certain conditions on $a$, $b$, and the $\mathfrak{b}$-module structure of $W$ are satisfied.
• Two such simple modules $\mathcal{L}(W,\lambda,a,b)$ and $\mathcal{L}(W',\lambda',a',b')$ are isomorphic if and only if $\lambda = \lambda'$, $a = a'$, $b = b'$, and $W \cong W'$ as $\mathfrak{W}$-modules.
• When $\lambda = 1$, the modules $\mathcal{L}(W,1,a,b)$ are not simple, and all their submodules are completely determined via the structure of $W$ and the action of $\mathfrak{b}$.
• The modules $\mathcal{L}(W,\lambda,a,b)$ include all modules from [CM] as special cases when $W$ is a specific highest weight $\mathfrak{W}$-module.
• The constructed modules are not isomorphic to any simple submodules of $V(\dot{c},h) \otimes A'_{a_1,b_1}$, as shown by the non-vanishing action of the operator $X_{l,m}$ on their weight vectors.
• The operator $X_{l,m}$ acts nontrivially on weight vectors in $\mathcal{L}(W,\lambda,a,b)$ for $\lambda \neq 1$, while it vanishes on vectors in tensor product modules, proving non-isomorphism.