Vertex operators for imaginary $\mathfrak{gl}_2$ subalgebras in the Monster Lie Algebra
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911785451257856 |
|---|---|
| author | Addabbo, Darlayne Carbone, Lisa Jurisich, Elizabeth Khaqan, Maryam Murray, Scott H. |
| author_facet | Addabbo, Darlayne Carbone, Lisa Jurisich, Elizabeth Khaqan, Maryam Murray, Scott H. |
| contents | The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space of the vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is the vertex algebra corresponding to the rank 2 even unimodular lattice $\textrm{II}_{1,1}$. We construct vertex algebra elements that project to bases for subalgebras of $\mathfrak m$ isomorphic to $\mathfrak{gl}_{2}$, corresponding to each imaginary simple root, denoted $(1,j)$ for $j>0$. Our method requires the existence of pairs of primary vectors in $V^{\natural}$ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group $\mathbb{M}$ on the subspace of primary vectors in $V^\natural$ induces an $\mathbb{M}$-action on the set of $\mathfrak{gl}_2$ subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of $V^\natural$ to prove that this action is non-trivial for small values of $j$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16178 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Vertex operators for imaginary $\mathfrak{gl}_2$ subalgebras in the Monster Lie Algebra Addabbo, Darlayne Carbone, Lisa Jurisich, Elizabeth Khaqan, Maryam Murray, Scott H. Representation Theory Mathematical Physics 17B69, 17B67, 20D08 The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space of the vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is the vertex algebra corresponding to the rank 2 even unimodular lattice $\textrm{II}_{1,1}$. We construct vertex algebra elements that project to bases for subalgebras of $\mathfrak m$ isomorphic to $\mathfrak{gl}_{2}$, corresponding to each imaginary simple root, denoted $(1,j)$ for $j>0$. Our method requires the existence of pairs of primary vectors in $V^{\natural}$ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group $\mathbb{M}$ on the subspace of primary vectors in $V^\natural$ induces an $\mathbb{M}$-action on the set of $\mathfrak{gl}_2$ subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of $V^\natural$ to prove that this action is non-trivial for small values of $j$. |
| title | Vertex operators for imaginary $\mathfrak{gl}_2$ subalgebras in the Monster Lie Algebra |
| topic | Representation Theory Mathematical Physics 17B69, 17B67, 20D08 |
| url | https://arxiv.org/abs/2210.16178 |