HEXAGONbeta

Multiplicative bases and deformations of representation-finite algebras

Primarily AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.RA — Rings and Algebras

Contributed by Haruhisa Enomoto ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

Bautista, Gabriel, Roiter, and Salmer\'on proved that every finite-dimensional representation-finite algebra over an algebraically closed field admits a multiplicative basis, in which products of basis elements are zero or basis elements. Outside characteristic two, they also proved that such an algebra is standard: its category of indecomposable modules is presented by its Auslander--Reiten quiver and mesh relations. We give another proof of both theorems. For a basic algebra, grading morphisms between indecomposable modules by powers of the categorical radical and restricting to projectives yields its standard form, a standard algebra with the same Auslander--Reiten quiver. After constructing a multiplicative basis of the standard form, we recover the original algebra by filtered deformation. Positive-degree Hochschild cohomology embeds into cokernels measuring errors in mesh relations. In characteristic two, lifting derivations modulo four produces cocycles supported on zero basis products; the resulting polynomial family contains the original algebra with coefficients zero or one.

Provenance statement

I instructed Astra to come up with proofs for classical results of BGRS: the existence of multiplicative basis of representation-finite algebras and mesh standardness for characteristic away from 2, then Astra successfully has found it using tau-category and mesh grading deformation. Astra and Opus 5.5 and Gemini 4 Argon revised paper a lot of times. Since this seems to be very important results about structure theory of representation-finite algebras which I think has lots of values to community of representation theorists, I am now going to understand the whole proof. While it will take several weeks or months, I decided to put it here. After I understand reasonably amount of the paper, I will probably put this in arXiv.

Formalizations

Tools used

OpenAI
CodexVersion GPT-6-Astra
Anthropic
Claude OpusVersion Claude Opus 5.5
Google
GeminiVersion Gemini 4 Argon

References

  1. M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras , Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995.
  2. R. Bautista, P. Gabriel, A. V. Roiter, and L. Salmerón, Representation-finite algebras and multiplicative bases , Invent. Math. 81 (1985), 217--285.
  3. J. Biał kowski, An algorithm for the periodicity of deformed preprojective algebras of Dynkin types E6E_6, E7E_7 and E8E_8 , Appl. Math. Comput. 409 (2021), article 126289.
  4. J. Biał kowski, K. Erdmann, and A. Skowroński, Deformed preprojective algebras of generalized Dynkin type , Trans. Amer. Math. Soc. 359 (2007), 2625--2650.
  5. K. Bongartz, On mild contours in ray categories , Algebr. Represent. Theory 17 (2014), 137--159.
  6. K. Bongartz, On representation-finite algebras and beyond , in Advances in representation theory of algebras , EMS Series of Congress Reports, European Mathematical Society, Zürich, 2014, 65--101.
  7. K. Bongartz and P. Gabriel, Covering spaces in representation-theory , Invent. Math. 65 (1982), 331--378.
  8. S. Brenner, M. C. R. Butler, and A. D. King, Periodic algebras which are almost Koszul , Algebr. Represent. Theory 5 (2002), 331--368.
  9. O. Bretscher and P. Gabriel, The standard form of a representation-finite algebra , Bull. Soc. Math. France 111 (1983), 21--40.
  10. R.-O. Buchweitz and S. Liu, Hochschild cohomology and representation-finite algebras , Proc. London Math. Soc. (3) 88 (2004), 355--380.
Show all 27 referencesShow fewer references
  1. T. Etgü and Y. Lekili, Koszul duality patterns in Floer theory , Geom. Topol. 21 (2017), 3313--3389.
  2. M. Gerstenhaber, On the deformation of rings and algebras , Ann. of Math. (2) 79 (1964), 59--103.
  3. M. Gerstenhaber and S. D. Schack, Relative Hochschild cohomology, rigid algebras, and the Bockstein , J. Pure Appl. Algebra 43 (1986), 53--74.
  4. O. Iyama, The relationship between homological properties and representation theoretic realization of artin algebras , Trans. Amer. Math. Soc. 357 (2005), 709--734.
  5. O. Iyama, τ\tau-Categories I: Ladders , Algebr. Represent. Theory 8 (2005), 297--321.
  6. J. P. Jans, On the indecomposable representations of algebras , Ann. of Math. (2) 66 (1957), 418--429.
  7. D. Kaplan, Multiplicative preprojective algebras of Dynkin quivers , J. Pure Appl. Algebra 227 (2023), article 107146.
  8. A. A. Kostin and B. V. Novikov, Semigroup cohomology as a derived functor , Filomat 15 (2001), 17--23.
  9. H. Kupisch, Symmetrische Algebren mit endlich vielen unzerlegbaren Darstellungen. I , J. Reine Angew. Math. 219 (1965), 1--25; II , J. Reine Angew. Math. 245 (1970), 1--14.
  10. Y. Lekili and K. Ueda, Homological mirror symmetry for Milnor fibers of simple singularities , Algebr. Geom. 8 (2021), 562--586.
  11. M. Linckelmann, Integrable derivations and stable equivalences of Morita type , Proc. Edinb. Math. Soc. (2) 61 (2018), 343--362.
  12. W. Lowen, Hochschild cohomology, the characteristic morphism and derived deformations , Compos. Math. 144 (2008), 1557--1580.
  13. C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück , Comment. Math. Helv. 55 (1980), 199--224.
  14. C. Riedtmann, Representation-finite selfinjective algebras of class DnD_n , Compositio Math. 49 (1983), 231--282.
  15. C. Riedtmann, On stable blocks of Auslander-algebras , Trans. Amer. Math. Soc. 283 (1984), 485--505.
  16. C. M. Ringel, Dieter Vossieck and the development of the representation theory of Artin algebras , lecture at the workshop Discrete Categories in Representation Theory , Bielefeld, 2018, arXiv:1808.02301 (https://arxiv.org/abs/1808.02301).arXiv
  17. T. Schedler, Zeroth Hochschild homology of preprojective algebras over the integers , Adv. Math. 299 (2016), 451--542.

Version history

  1. v1Submitted by Haruhisa EnomotoInitial depositCurrentOct 23, 2026