TY - JOUR

T1 - Cofiniteness conditions, projective covers and the logarithmic tensor product theory

AU - Huang, Yi Zhi

N1 - Funding Information:
The author gratefully acknowledges the partial support from NSF grant DMS-0401302. The author is grateful to Antun Milas and Jim Lepowsky for comments.

PY - 2009/4

Y1 - 2009/4

N2 - 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 C1-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 AN (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 also satisfies Condition 3 that irreducible V-modules are R-graded and C1-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under operations {squared falling diagonal slash}P (z) for z ∈ C×. 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 C2-cofinite, Conditions 1-3 are satisfied and thus all the conclusions hold.

AB - 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 C1-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 AN (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 also satisfies Condition 3 that irreducible V-modules are R-graded and C1-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under operations {squared falling diagonal slash}P (z) for z ∈ C×. 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 C2-cofinite, Conditions 1-3 are satisfied and thus all the conclusions hold.

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U2 - 10.1016/j.jpaa.2008.07.016

DO - 10.1016/j.jpaa.2008.07.016

M3 - Article

AN - SCOPUS:58149513975

VL - 213

SP - 458

EP - 475

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 4

ER -