## Strongly Holomorphic C |

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1-terms abelian Additionally affine Lie algebra algebra commutation relations basis element C2-cofinite calculations Cartan subalgebra central charge 24 compute the trace conformal vector contribution creation axiom decomposition define degree-four spherical harmonics Dong and Mason Eisenstein series equation 22 finite-dimensional following lemma following proposition form of weight graded dimension graded trace equal highest-weight vector holomorphic modular forms homogeneous degree-four spherical implies invariant bilinear form isomorphic Jacobi identity Killing form Leech lattice Lie algebra commutation linear combination main theorem Mason DM00 moonshine module Niemeier lattice non-zero cusp forms orthonormal basis polynomial Proof Proposition 3.3 Resz root systems s^-subalgebra Section simple Lie algebra square-bracket VOA strong CFT-type strongly holomorphic VOAs strongly rational VOA tr tr tr v o(v trace of o(v vacuum vector vector space vector with graded vertex operator algebras Virasoro weight vector weight-two Weyl group Zv+(v,q