Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cofiniteness conditions, projective covers and the logarithmic tensor
  product theory   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^{\times}. 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.

17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph