2015/10/31 by Matthew Pressland · 19 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bimodule #Calabi–Yau manifold #Cluster algebra #Endomorphism #Idempotence #Indecomposable module #Jacobian matrix and determinant #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum #math.RT #msc:13F60 #msc:16G20 #msc:16G50 #msc:18E30
paper · pdf · doi:10.1007/s00209-016-1837-0
published in Mathematische Zeitschrift 287(1-2), 555-585 (Springer Science+Business Media) · 35 pages, comments welcome. v3: final version, to appear in Math. Zeitschrift
openalex publication_date 2017/01/03 · arxiv created 2017/01/04 · arxiv updated 2017/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1) -cluster-tilting objects in certain stably (d-1) -Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1) -cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger ‘bimodule’ internally d-Calabi–Yau condition, this Frobenius category is stably (d-1) -Calabi–Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi–Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot’s construction in the coefficient-free case.