论文标题
分支规则分解级别1 $ e_8^{（1）} $ - 相对于不规则的subalgebra $ f_4^{（1）} \ oplus g_2^{（1）} $
Branching rule decomposition of the level-1 $E_8^{(1)}$-module with respect to the irregular subalgebra $F_4^{(1)} \oplus G_2^{(1)}$
论文作者
论文摘要
给定一个类型$ e_8 $的谎言代数，可以使用$ e_6 $和$ d_4 $ dynkin图的dynkin图自动形态，以找到$ f_4 \ oplus g_2 $的sibalgebra。这些自动形态可以将这些代数的仿射kac-moody对应物提升到类型的$ f_4^{（1）} \ oplus g_2^{（1）} $的sibergebra中。我们将考虑级别-1不可约的$ e_8^{（1）} $ - 模块$ v^{λ_0} $并研究其分支规则，这就是它将其直接分解为不可约的$ f_4^{（1）} \ oplus g_2^{（1）} $ - 模块。 我们使用使用theta函数和所谓的“字符串函数”的KAC-Peterson的字符公式来计算这些分支规则。我们将在计算中利用Jacobi的Ramanujan和Borweins的Theta函数（及其各自的属性和身份），包括涉及Rogers-Ramanujan系列的一些身份。 Virasoro角色理论用于验证KAC和Peterson所说的字符串功能。我们还研究了一些有趣的$η$ Quortients的解剖。
Given a Lie algebra of type $E_8$, one can use Dynkin diagram automorphisms of the $E_6$ and $D_4$ Dynkin diagrams to locate a subalgebra of type $F_4\oplus G_2$. These automorphisms can be lifted to the affine Kac-Moody counterparts of these algebras and give a subalgebra of type $F_4^{(1)}\oplus G_2^{(1)}$ within a type $E_8^{(1)}$ Kac-Moody Lie algebra. We will consider the level-1 irreducible $E_8^{(1)}$-module $V^{Λ_0}$ and investigate its branching rule, that is how it decomposes as a direct sum of irreducible $F_4^{(1)}\oplus G_2^{(1)}$-modules. We calculate these branching rules using a character formula of Kac-Peterson which uses theta functions and the so-called "string functions." We will make use of Jacobi's, Ramanujan's and the Borweins' theta functions (and their respective properties and identities) in our calculation, including some identities involving the Rogers-Ramanujan series. Virasoro character theory is used to verify string functions stated by Kac and Peterson. We also investigate dissections of some interesting $η$-quotients.
