Fusion Rules for the Lattice Vertex Operator Algebra $V_L$

For a positive-definite, even, integral lattice $L$, the lattice vertex operator algebra $V_L$ is known to be rational and $C_2$-cofinite, and thus the fusion products of its modules always exist. The fusion product of two untwisted irreducible $V_L$-modules is well-known, namely $V_{L+\lambda} \boxtimes_{V_L} V_{L+\mu} = V_{L + \lambda + \mu}$. In this paper, we determine the other two fusion products: $V_{L+\lambda} \boxtimes_{V_L} V_L^{T_{\chi}}$ and $V_L^{T_{\chi_1}} \boxtimes_{V_L} V_L^{T_{\chi_2}}$.