On $\mathbb{N}$-graded vertex algebras associated with Gorenstein algebras
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910739169542144 |
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| author | Keene, Alex Soltermann, Christian Yamskulna, Gaywalee |
| author_facet | Keene, Alex Soltermann, Christian Yamskulna, Gaywalee |
| contents | This paper investigates the algebraic structure of indecomposable $\mathbb{N}$-graded vertex algebras $V = \bigoplus_{n=0}^{\infty} V_n$, emphasizing the intricate interactions between the commutative associative algebra $V_0$, the Leibniz algebra $V_1$ and how non-degenerate bilinear forms on $V_0$ influence their overall structure. We establish foundational properties for indecomposability and locality in $\mathbb{N}$-graded vertex algebras, with our main result demonstrating the equivalence of locality, indecomposability, and specific structural conditions on semiconformal-vertex algebras. The study of symmetric invariant bilinear forms of semiconformal-vertex algebra is investigated. We also examine the structural characteristics of $V_0$ and $V_1$, demonstrating conditions under which certain $\mathbb{N}$-graded vertex algebras cannot be quasi vertex operator algebras, semiconformal-vertex algebras, or vertex operator algebras, and explore $\mathbb{N}$-graded vertex algebras $V=\bigoplus_{n=0}^{\infty}V_n$ associated with Gorenstein algebras. Our analysis includes examining the socle, Poincaré duality properties, and invariant bilinear forms of $V_0$ and their influence on $V_1$, providing conditions for embedding rank-one Heisenberg vertex operator algebras within $V$. Supporting examples and detailed theoretical insights further illustrate these algebraic structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $\mathbb{N}$-graded vertex algebras associated with Gorenstein algebras Keene, Alex Soltermann, Christian Yamskulna, Gaywalee Quantum Algebra Rings and Algebras Representation Theory 17B69 This paper investigates the algebraic structure of indecomposable $\mathbb{N}$-graded vertex algebras $V = \bigoplus_{n=0}^{\infty} V_n$, emphasizing the intricate interactions between the commutative associative algebra $V_0$, the Leibniz algebra $V_1$ and how non-degenerate bilinear forms on $V_0$ influence their overall structure. We establish foundational properties for indecomposability and locality in $\mathbb{N}$-graded vertex algebras, with our main result demonstrating the equivalence of locality, indecomposability, and specific structural conditions on semiconformal-vertex algebras. The study of symmetric invariant bilinear forms of semiconformal-vertex algebra is investigated. We also examine the structural characteristics of $V_0$ and $V_1$, demonstrating conditions under which certain $\mathbb{N}$-graded vertex algebras cannot be quasi vertex operator algebras, semiconformal-vertex algebras, or vertex operator algebras, and explore $\mathbb{N}$-graded vertex algebras $V=\bigoplus_{n=0}^{\infty}V_n$ associated with Gorenstein algebras. Our analysis includes examining the socle, Poincaré duality properties, and invariant bilinear forms of $V_0$ and their influence on $V_1$, providing conditions for embedding rank-one Heisenberg vertex operator algebras within $V$. Supporting examples and detailed theoretical insights further illustrate these algebraic structures. |
| title | On $\mathbb{N}$-graded vertex algebras associated with Gorenstein algebras |
| topic | Quantum Algebra Rings and Algebras Representation Theory 17B69 |
| url | https://arxiv.org/abs/2412.07918 |