[Paper Review] The Triplet Vertex Operator Algebra W(p) and the Restricted Quantum Group at Root of Unity

Kiyokazu Nagatomo, Akihiro Tsuchiya|ArXiv.org|Feb 26, 2009

Algebraic structures and combinatorial models20 references66 citations

TL;DR

This paper establishes a categorical equivalence between the abelian category of modules over the triplet vertex operator algebra $W(p)$ and the category of finite-dimensional modules over the restricted quantum group $¯{U}_q(sl_2)$ at $q = e^{\pi i/p}$, proving a long-standing conjecture by Feigin et al. The authors construct projective covers $\mathcal{P}_s^\pm$ for simple $W(p)$-modules $\mathcal{X}_s^\pm$, analyze their structure via screening operators, and use this to show that $W(p)$-mod is equivalent to $\u00af{U}_q(sl_2)$-mod as abelian categories, with blocks indexed by $s = 0, \dots, p$, and non-semisimple structure for $1 \leq s \leq p-1$. This provides a Kazhdan-Lusztig-type correspondence in logarithmic conformal field theory.

ABSTRACT

We prove the abelian category of the modules over triplet VOA W(p) is category equivalent to the abelian category of the modules over quantum algebra of type sl_2 at root of unity.

Motivation & Objective

• To resolve the conjecture by Feigin et al. that the abelian category $W(p)$-mod is equivalent to the category of finite-dimensional modules for the restricted quantum group $\u00af{U}_q(sl_2)$ at $q = e^{\pi i/p}$.
• To construct and analyze the projective covers $\mathcal{P}_s^\pm$ of the simple $W(p)$-modules $\mathcal{X}_s^\pm$ for $1 \leq s \leq p$.
• To determine the structure of $W(p)$-mod via block decomposition, showing non-semisimplicity for $1 \leq s \leq p-1$ and semisimplicity for $s = 0, p$.
• To compute Ext$^1$ groups between simple objects and establish the endomorphism algebra $B_s$ of the projective cover in each block.

Proposed method

• Construct $W(p)$-modules $\mathcal{P}_s^\pm$ using iterated integrals of screening operators $Q_-^{[d_s^\varepsilon]}(z)$, generalizing the method of Fjelsted et al.
• Use intertwining operators from $Q_+(z)$ and $Q_-^{[d_s^\varepsilon]}(z)$ to analyze Fock space representations and module structures.
• Prove that $\mathcal{P}_s^\pm$ are projective covers of $\mathcal{X}_s^\pm$ by showing vanishing of Ext$^1$ groups and non-diagonalizability of $T(0)$.
• Determine Zhu’s algebra $A_0(W(p))$ via the structure of $\mathcal{P}_s^\pm$, leading to block decomposition of $W(p)$-mod.
• Show that the endomorphism algebra $B_s = \mathrm{End}_{C_s}(\mathcal{P}_s^+ \oplus \mathcal{P}_s^-)$ is isomorphic to $B(\bar{U})$, the endomorphism algebra of the projective cover in $\bar{U}_q(sl_2)$-mod.
• Apply the equivalence theorem (Proposition 6-3) to conclude that $W(p)$-mod and $\bar{U}_q(sl_2)$-mod are equivalent abelian categories.

Experimental results

Research questions

• RQ1 Is there a categorical equivalence between the abelian category $W(p)$-mod and the category of finite-dimensional modules for the restricted quantum group $\bar{U}_q(sl_2)$ at $q = e^{\pi i/p}$?
• RQ2 What is the structure of the projective covers $\mathcal{P}_s^\pm$ of the simple $W(p)$-modules $\mathcal{X}_s^\pm$?
• RQ3 How does the block decomposition of $W(p)$-mod compare to that of $\bar{U}_q(sl_2)$-mod, particularly in terms of semisimplicity and Ext$^1$ groups?
• RQ4 Can the endomorphism algebra $B_s$ of the projective cover in each block of $W(p)$-mod be identified with the known algebra $B(\bar{U})$ from quantum group representation theory?
• RQ5 What is the maximal length of Jordan blocks for the action of $T(0)$ on indecomposable $W(p)$-modules?

Key findings

• The abelian category $W(p)$-mod admits a block decomposition $\bigoplus_{s=0}^p C_s$, where $C_0$ and $C_p$ are semisimple with one simple object each, and $C_s$ for $1 \leq s \leq p-1$ has two simple objects $\mathcal{X}_s^+$ and $\mathcal{X}_s^-$.
• The projective covers $\mathcal{P}_s^\pm$ of $\mathcal{X}_s^\pm$ are self-dual and injective, and $\mathcal{P}_s^\pm$ is a projective cover of $\mathcal{X}_s^\pm$ for $1 \leq s \leq p-1$.
• The Ext$^1$ groups between simple objects in $C_s$ are nontrivial for $1 \leq s \leq p-1$, confirming that $C_s$ is not semisimple.
• The endomorphism algebra $B_s = \mathrm{End}_{C_s}(\mathcal{P}_s^+ \oplus \mathcal{P}_s^-)$ is isomorphic to $B(\bar{U})$, the 8-dimensional algebra of endomorphisms of the projective cover in $\bar{U}_q(sl_2)$-mod.
• The main result is the equivalence of abelian categories: $W(p)$-mod $\simeq \bar{U}_q(sl_2)$-mod, as established via the equivalence of their block categories and isomorphic endomorphism algebras.
• The length $l(M)$ of any $W(p)$-module $M$ satisfies $l(M) \leq 1$, and any indecomposable module with $l(M) = 1$ is isomorphic to $\mathcal{P}_s^\pm$ for some $1 \leq s \leq p-1$.