Structure and representations of the coset vertex operator algebra $C(L_{\widehat{osp(1\mathrm{\mid }2)}}(2,0),L_{\widehat{osp(1\mathrm{\mid }2)}}(1,0{)}^{\otimes 2})$

In this paper, we determinate the structure of the coset vertex operator algebra $C(L_{\widehat{osp(1\mathrm{\mid }2)}}(2,0),L_{\widehat{osp(1\mathrm{\mid }2)}}(1,0{)}^{\otimes 2})$. We prove that $C(L_{\widehat{osp(1\mathrm{\mid }2)}}(2,0),L_{\widehat{osp(1\mathrm{\mid }2)}}(1,0{)}^{\otimes 2})$ is an extension of the rational vertex operator algebra $L(c_{10,7},0)$. Representations and fusion rules for $C(L_{\widehat{osp(1\mathrm{\mid }2)}}(2,0),L_{\widehat{osp(1\mathrm{\mid }2)}}(1,0{)}^{\otimes 2})$ are also completely determined.