Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2206.00163 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917508507762688 |
|---|---|
| author | Carey, Joshua D. |
| author_facet | Carey, Joshua D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_00163 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Branching rule decomposition of the level-1 $E_8^{(1)}$-module with respect to the irregular subalgebra $F_4^{(1)} \oplus G_2^{(1)}$ Carey, Joshua D. Representation Theory 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. |
| title | Branching rule decomposition of the level-1 $E_8^{(1)}$-module with respect to the irregular subalgebra $F_4^{(1)} \oplus G_2^{(1)}$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2206.00163 |