Normed representations of weight quivers
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909689218859008 |
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| author | Liu, Yu-Zhe |
| author_facet | Liu, Yu-Zhe |
| contents | Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\mathscr{A}^p_ς$ in this paper whose object is a triple $(N,v,δ)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $δ: N^{\oplus_p 2^{\dim A}} \to N$ and each morphism $(N,v,δ) \to (N',v',δ')$ is given by an $(A,B)$-homomorphism $θ: N\to N'$ such that $θ(v)=v'$ and $δ' θ^{\oplus 2^{\dim A}} = θδ$ hold. We show that $\mathscr{A}^p_ς$ has an initial object such that Daniell integration, Bochner integration, Lebesgue integration, Stone--Weierstrass Approximation Theorem, power series expansion, and Fourier series expansion are morphisms in $\mathscr{A}^p_ς$ starting with this initial object. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normed representations of weight quivers Liu, Yu-Zhe Representation Theory Category Theory Functional Analysis 16G10, 46M40 Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\mathscr{A}^p_ς$ in this paper whose object is a triple $(N,v,δ)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $δ: N^{\oplus_p 2^{\dim A}} \to N$ and each morphism $(N,v,δ) \to (N',v',δ')$ is given by an $(A,B)$-homomorphism $θ: N\to N'$ such that $θ(v)=v'$ and $δ' θ^{\oplus 2^{\dim A}} = θδ$ hold. We show that $\mathscr{A}^p_ς$ has an initial object such that Daniell integration, Bochner integration, Lebesgue integration, Stone--Weierstrass Approximation Theorem, power series expansion, and Fourier series expansion are morphisms in $\mathscr{A}^p_ς$ starting with this initial object. |
| title | Normed representations of weight quivers |
| topic | Representation Theory Category Theory Functional Analysis 16G10, 46M40 |
| url | https://arxiv.org/abs/2507.06962 |