In this paper, under the assumption that the diagonal coset vertex operator algebra $$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) is rational and $$C_2$$ C 2 -cofinite, the global dimension of $$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) is obtained and the quantum dimensions of multiplicity spaces viewed as $$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) -modules are also obtained. As an application, a method to classify irreducible modules of $$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) is provided. As an example, we prove that the diagonal coset vertex operator algebra $$C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$$ C ( L E 8 ( k + 2 , 0 ) , L E 8 ( k , 0 ) ⊗ L E 8 ( 2 , 0 ) ) is rational, $$C_2$$ C 2 -cofinite, and classify irreducible modules of $$C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$$ C ( L E 8 ( k + 2 , 0 ) , L E 8 ( k , 0 ) ⊗ L E 8 ( 2 , 0 ) ) .

中文翻译：

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一些对角陪集顶点算子代数的量子维数和不可约模

本文假设对角陪集顶点算子代数$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_ {\mathfrak {g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) 是有理数且 $$C_2$ $C 2 -cofinite，$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak { g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) 被获得，多重空间的量子维数被视为$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$ $C (L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) -模也得到了。作为应用，一种对$$C(L_{\mathfrak {g}}(k+l,0),L_{\mathfrak {g}}(k,0)\otimes L_{\mathfrak {g}}(l,0))$$ C ( L g ( k + l , 0 ) , L g ( k , 0 ) ⊗ L g ( l , 0 ) ) 被提供。例如，我们证明对角陪集顶点算子代数 $$C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0)) $$ C ( LE 8 ( k + 2 , 0 ) , LE 8 ( k , 0 ) ⊗ LE 8 ( 2 , 0 ) ) 是有理的， $$C_2$$ C 2 -cofinite，并对$$的不可约模进行分类C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$$ C ( LE 8 ( k + 2 , 0 ) , LE 8 (k, 0) ⊗ LE 8 (2, 0))。