Pseudotraces on Almost Unital and Finite-Dimensional Algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916005116116992 |
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| author | Gui, Bin Zhang, Hao |
| author_facet | Gui, Bin Zhang, Hao |
| contents | We introduce the notion of almost unital and finite-dimensional (AUF) algebras, which are associative $\mathbb C$-algebras that may be non-unital or infinite-dimensional, but have sufficiently many idempotents. We show that the pseudotrace construction, originally introduced by Hattori and Stallings for unital finite-dimensional algebras, can be generalized to AUF algebras.
Let $A$ be an AUF algebra. Suppose that $G$ is a projective generator in the category $\mathrm{Coh}_{\mathrm{L}}(A)$ of finitely generated left $A$-modules that are quotients of free left $A$-modules, and let $B = \mathrm{End}_{A,-}(G)^{\mathrm{opp}}$. We prove that the pseudotrace construction yields an isomorphism between the spaces of symmetric linear functionals $\mathrm{SLF}(A)\xrightarrow{\simeq} \mathrm{SLF}(B)$, and that the non-degeneracies on the two sides are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudotraces on Almost Unital and Finite-Dimensional Algebras Gui, Bin Zhang, Hao Quantum Algebra Rings and Algebras Representation Theory We introduce the notion of almost unital and finite-dimensional (AUF) algebras, which are associative $\mathbb C$-algebras that may be non-unital or infinite-dimensional, but have sufficiently many idempotents. We show that the pseudotrace construction, originally introduced by Hattori and Stallings for unital finite-dimensional algebras, can be generalized to AUF algebras. Let $A$ be an AUF algebra. Suppose that $G$ is a projective generator in the category $\mathrm{Coh}_{\mathrm{L}}(A)$ of finitely generated left $A$-modules that are quotients of free left $A$-modules, and let $B = \mathrm{End}_{A,-}(G)^{\mathrm{opp}}$. We prove that the pseudotrace construction yields an isomorphism between the spaces of symmetric linear functionals $\mathrm{SLF}(A)\xrightarrow{\simeq} \mathrm{SLF}(B)$, and that the non-degeneracies on the two sides are equivalent. |
| title | Pseudotraces on Almost Unital and Finite-Dimensional Algebras |
| topic | Quantum Algebra Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2508.00431 |