[Paper Review] Extending Representations of sl(2) to Witt and Virasoro algebras

Francisco J. Plaza Martín, Carlos Prieto|arXiv (Cornell University)|Nov 20, 2014

Algebraic structures and combinatorial models10 references4 citations

TL;DR

This paper provides a complete characterization of when irreducible weight modules for $σ\mathfrak{sl}(2)$ extend to representations of the Witt and Virasoro algebras. Using a step-by-step extension strategy through subalgebras ${\operatorname{\bf Witt}}_{>}$, ${\operatorname{\bf Witt}}_{<}$, and ${\operatorname{\bf Witt}}$, the authors derive explicit formulas for compatible actions of $L_i$ on Verma and dual Verma modules, showing that simple Verma modules $M(\lambda)$ admit exactly two $\operatorname{\bf Vir}$-module structures for $\lambda \neq -1,-2$, and one otherwise, with central charge zero.

ABSTRACT

We study when an sl(2)-representation extends to a representation of the Witt and Virasoro algebras. We give a criterion for extendability and apply it to certain classes of weight sl(2)-modules. For all simple weight sl(2)-modules and those in any of the abelian Krull-Schmidt categories of weight modules whose unique simple object is a dense module, we fully characterize which ones admit extensions, and we obtain explicit expressions for all of them. We also give partial results in the same direction for the abelian categories of weight modules which have two and three simple objects.

Motivation & Objective

• To determine when a representation of $\mathfrak{sl}(2)$ extends to a representation of the Witt and Virasoro algebras.
• To classify all possible extensions of simple weight $\mathfrak{sl}(2)$-modules to $\operatorname{\bf Vir}$, particularly Verma and dual Verma modules.
• To analyze the obstruction structure in the extension process through intermediate subalgebras ${\operatorname{\bf Witt}}_{>}$ and ${\operatorname{\bf Witt}}_{<}$.
• To provide explicit realizations of extended actions using hypergeometric-type functions $P(\cdot, \cdot)$.

Proposed method

• The authors use a step-by-step extension strategy, analyzing the inclusion chain $\mathfrak{sl}(2) \subset {\operatorname{\bf Witt}}_{>} \subset {\operatorname{\bf Witt}} \subset \operatorname{\bf Vir}$.
• They derive necessary and sufficient conditions for extending an $\mathfrak{sl}(2)$-representation to ${\operatorname{\bf Witt}}_{>}$ by solving a system of equations involving matrix coefficients.
• Explicit formulas for the action of $L_i$ on weight vectors are constructed using the hypergeometric-type function $P(a,b)$, parameterized by $\lambda$ and $j$.
• The compatibility between ${\operatorname{\bf Witt}}_{>}$ and ${\operatorname{\bf Witt}}_{<}$ structures is checked to determine extendability to the full Witt algebra.
• The central charge $C$ is set to zero in all constructed Virasoro extensions, and the highest/lowest weight nature of the modules is verified.

Experimental results

Research questions

• RQ1 For which simple weight $\mathfrak{sl}(2)$-modules does there exist a compatible $\operatorname{\bf Vir}$-module structure?
• RQ2 What are the explicit formulas for the action of $L_i$ on the weight vectors of Verma modules under such extensions?
• RQ3 How many distinct $\operatorname{\bf Vir}$-module structures can a given $\mathfrak{sl}(2)$-module admit?
• RQ4 What are the obstructions to extending $\mathfrak{sl}(2)$-representations to $\operatorname{\bf Vir}$, and how do they arise from intermediate subalgebras?
• RQ5 How do the extension properties differ for Verma modules $M(\lambda)$ and dual Verma modules $\bar{M}(\lambda+2)$?

Key findings

• The Verma module $M(\lambda)$ admits exactly two compatible ${\operatorname{\bf Witt}}_{<}$-module structures for $\lambda \neq 1,2,\ldots$, given by $\rho_{<}^{\pm}(L_i)$ with explicit formulas involving $P(\cdot, \cdot)$.
• For $\lambda = -1,-2$, the two structures coincide, resulting in a unique extension to ${\operatorname{\bf Witt}}_{<}$.
• The Verma module $M(\lambda)$ admits a unique compatible ${\operatorname{\bf Witt}}$-module structure given by $\rho_{<}^{-}(L_i)$, and this extends uniquely to a $\operatorname{\bf Vir}$-module with $\rho_{<}^{-}(C) = 0$.
• The $\operatorname{\bf Vir}$-module structure on $M(\lambda)$ is a highest weight module of highest weight $(0, -\frac{1}{2}\lambda)$.
• The dual Verma module $\bar{M}(\lambda+2)$ admits a unique compatible ${\operatorname{\bf Witt}}$-module structure via $\rho_{<}^{-}(L_i)$, and extends uniquely to a $\operatorname{\bf Vir}$-module with $\rho_{<}^{-}(C) = 0$, forming a lowest weight module of lowest weight $(0, -\frac{1}{2}(\lambda+2))$.
• In the case of three simple objects ($\tau = n^2$), the module ${\mathbf{V}}^{(n)}$ admits no nontrivial extension to ${\operatorname{\bf Witt}}_{>}$, ${\operatorname{\bf Witt}}$, or $\operatorname{\bf Vir}$, while extensions exist for $M(-n-1)$ and $\bar{M}(n+1)$.