[Paper Review] Cofiniteness conditions, projective covers and the logarithmic tensor product theory

Yi-Zhi Huang|ArXiv.org|Dec 26, 2007

Algebraic structures and combinatorial models35 references3 citations

TL;DR

This paper establishes the existence of projective covers for irreducible modules in the category of grading-restricted generalized modules over a vertex operator algebra $V$ under $C_1$-cofiniteness and finite-dimensionality conditions on $A_N(V)$. The key result is that the category becomes a finite abelian category and, under additional $R$-grading and $C_1$-cofiniteness, is closed under the logarithmic tensor product operations, forming a braided tensor category.

ABSTRACT

We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C_{1}-cofinite in the sense of Li. 2. There exists a positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_{N}(V) is finite dimensional. This result shows that the category of grading-restricted generalized V-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V satisfies in addition Condition 3, that irreducible V-modules are R-graded and C_{1}-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under P(z)-tensor product operations for z in C^{ imes}. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when $V$ is of positive energy and C_{2}-cofinite, Conditions 1--3 are satisfied and thus all the conclusions hold.

Motivation & Objective

• To prove the existence of projective covers for irreducible $V$-modules in the category of grading-restricted generalized $V$-modules.
• To show that under $C_1$-cofiniteness and finite-dimensionality of $A_N(V)$, the category becomes a finite abelian category over $\mathbb{C}$.
• To establish closure of the category under the logarithmic tensor product operations when $V$-modules are $\mathbb{R}$-graded and $C_1$-cofinite in the author's sense.
• To demonstrate that the resulting category equipped with the logarithmic tensor product structure is a braided tensor category.
• To extend the framework of logarithmic tensor product theory to include projective covers and finite abelian structure.

Proposed method

• Utilizes the $C_1$-cofiniteness condition of Li for the vertex operator algebra $V$ to ensure finiteness properties in the module category.
• Applies the finite-dimensionality of the associative algebra $A_N(V)$ for some positive integer $N$ to control the structure of irreducible modules.
• Constructs projective covers using the theory of grading-restricted generalized modules and the representation theory of $V$.
• Employs the logarithmic tensor product theory developed in [HLZ2], particularly the operations $\otimes_{P(1)}^\prime$, to define tensor structures.
• Relies on the existence of projective covers to ensure the category is closed under the logarithmic tensor product operations.
• Verifies that the category satisfies the axioms of a braided tensor category, including associativity, braiding, and unit isomorphisms, using the framework from [HLZ2].

Experimental results

Research questions

• RQ1 Under what conditions does the category of grading-restricted generalized $V$-modules admit projective covers for irreducible modules?
• RQ2 When is the category of grading-restricted generalized $V$-modules a finite abelian category over $\mathbb{C}$?
• RQ3 What conditions ensure that the category is closed under the logarithmic tensor product operations $\otimes_{P(1)}^\prime$?
• RQ4 How does the $\mathbb{R}$-grading and $C_1$-cofiniteness in the author's sense affect the tensor category structure?
• RQ5 Can the logarithmic tensor product theory be extended to yield a braided tensor category when projective covers exist?

Key findings

• Projective covers exist for all irreducible $V$-modules in the category of grading-restricted generalized $V$-modules when $V$ is $C_1$-cofinite and $A_N(V)$ is finite-dimensional.
• The category of grading-restricted generalized $V$-modules becomes a finite abelian category over $\mathbb{C}$ under the stated cofiniteness and finiteness conditions.
• When $V$-modules are $\mathbb{R}$-graded and $C_1$-cofinite in the author's sense, the category is closed under the logarithmic tensor product $\otimes_{P(1)}^\prime$.
• The category equipped with the logarithmic tensor product structure, unit object $V$, braiding, and associativity isomorphisms from [HLZ2], forms a braided tensor category.
• The existence of projective covers ensures that the category supports the full structure of a braided tensor category in the logarithmic setting.
• The finite-dimensionality of $A_N(V)$ and bounded differences in lowest conformal weights by $N$ are essential for the finiteness and abelian structure of the category.