Cotype zeta functions enumerating subalgebras of $R$-algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916966289113088 |
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| author | Lee, Seok Hyeong Lee, Seungjai |
| author_facet | Lee, Seok Hyeong Lee, Seungjai |
| contents | We introduce and study subalgebra cotype zeta functions, multivariate zeta functions enumerating fixed-index subalgebras of $R$-algebras of a given cotype. This generalizes and unifies previous works on subalgebra zeta functions and cotype zeta functions of $R$-algebras. We prove the local functional equations for the generic Euler factors of these zeta functions, and give an explicit formula for the subalgebra cotype zeta function of a general $\mathbb{Z}$-Lie algebra $L$ of rank 3. We also give an asymptotic formula for the number of subalgebras $Λ$ of $L$ of index at most $X$ for which $L/Λ$ has rank at most, answering a question of Chinta, Kaplan, and Koplewitz. In particular, we show that unlike $\mathbb{Z}^3$, $\mathbb{Z}$-Lie algebras of rank 3 with additional multiplication structure exhibit different distribution of cocyclic subalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cotype zeta functions enumerating subalgebras of $R$-algebras Lee, Seok Hyeong Lee, Seungjai Rings and Algebras 11M41, 17B30, 20E07 We introduce and study subalgebra cotype zeta functions, multivariate zeta functions enumerating fixed-index subalgebras of $R$-algebras of a given cotype. This generalizes and unifies previous works on subalgebra zeta functions and cotype zeta functions of $R$-algebras. We prove the local functional equations for the generic Euler factors of these zeta functions, and give an explicit formula for the subalgebra cotype zeta function of a general $\mathbb{Z}$-Lie algebra $L$ of rank 3. We also give an asymptotic formula for the number of subalgebras $Λ$ of $L$ of index at most $X$ for which $L/Λ$ has rank at most, answering a question of Chinta, Kaplan, and Koplewitz. In particular, we show that unlike $\mathbb{Z}^3$, $\mathbb{Z}$-Lie algebras of rank 3 with additional multiplication structure exhibit different distribution of cocyclic subalgebras. |
| title | Cotype zeta functions enumerating subalgebras of $R$-algebras |
| topic | Rings and Algebras 11M41, 17B30, 20E07 |
| url | https://arxiv.org/abs/2504.18908 |