Subalgebras of lattice vertex operator algebras

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The structure of vertex operator algebra associated with a lattice L which is √<2> times an ordinary root lattice of type A, D, or E was studied. In 2001, the decomposition of the lattice vertex operator algebra into a direct sum of irreducible modules for a subalgebra generated by mutually orthogonal conformal vectors was studied.1. In the case L = √<2A>_1, all highest weight vectors of weight at most 2 are determined.2. In the case L= √<2D>_1, all highest weight vectors are determined for a certain choice of conformal vectors.3. Using the case L = √<2A>_3, a new type of vertex operator algebras are constructed from Z_8 codes.4. A complete decomposition of the moonshine module as a module for a certain vertex operator algebra related to the case L = √<2A>_3 is obtained.In 2002, the orbifold for an automorphism of order 3 was studied.5. The W_3 algebra of central charge 6/5 is discovered as a subalgebra of the orbifold of the lattice vertex operator algebra of type √<2A>_2.