A characterization of the rational vertex operator algebra   $V_{\Z\alpha}^{+}$}: II
  A characterization of vertex operator algebra $V_L^+$ for any rank one positive definite even lattice $L$ is given in terms of dimensions of homogeneous subspaces of small weights. This result reduces the classification of rational vertex operator algebras of central charge 1 to the characterization of three vertex operator algebras in the $E$-series of central charge one. 
Chongying Dong and Cuipo Jiang
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