Quantum dimensions and irreducible modules of some diagonal coset vertex operator algebras

In this paper, under the assumption that the diagonal coset vertex operator algebra C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$\end{document} is rational and C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_2$$\end{document}-cofinite, the global dimension of C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$\end{document} is obtained and the quantum dimensions of multiplicity spaces viewed as C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$\end{document}-modules are also obtained. As an application, a method to classify irreducible modules of C(Lg(k+l,0),Lg(k,0)⊗Lg(l,0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$\end{document} is provided. As an example, we prove that the diagonal coset vertex operator algebra C(LE8(k+2,0),LE8(k,0)⊗LE8(2,0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$$\end{document} is rational, C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_2$$\end{document}-cofinite, and classify irreducible modules of C(LE8(k+2,0),LE8(k,0)⊗LE8(2,0))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$$\end{document}.