Research ArticleNo Access

Algebraic properties of face algebras

Fabio Calderón

Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia

E-mail Address: facalderonm@unal.edu.co

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and

Chelsea Walton

Department of Mathematics, Rice University, Houston, TX 77005, USA

E-mail Address: notlaw@rice.edu

Corresponding author.

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https://doi.org/10.1142/S0219498823500767Cited by:0 (Source: Crossref)

Abstract

Prompted by an inquiry of Manin on whether a coacting Hopf-type structure $H$ $H$ and an algebra $A$ $A$ that is coacted upon share algebraic properties, we study the particular case of $A$ $A$ being a path algebra $𝕂 Q$ of a finite quiver $Q$ and $H$ being Hayashi’s face algebra $𝔥$ attached to $Q$ . This is motivated by the work of Huang, Wicks, Won and the second author, where it was established that the weak bialgebra coacting universally on $𝕂 Q$ (either from the left, right, or both sides compatibly) is $𝔥$ . For our study, we define the Kronecker square $\hat{Q}$ of $Q$ , and show that $𝔥 ≅ 𝕂 \hat{Q}$ as unital graded algebras. Then we obtain ring-theoretic and homological properties of $𝔥$ in terms of graph-theoretic properties of $Q$ by way of $\hat{Q}$ .

Communicated by L. H. Rowen

Keywords:

AMSC: 16T20, 05C25

References

1. M. Artin, W. Schelter and J. Tate, Quantum deformations of ${GL}_{n}$ , Commun. Pure Appl. Math. 44(8–9) (1991) 879–895. Crossref, Web of Science, Google Scholar
2. I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory, London Mathematical Society Student Texts, Vol. 65 (Cambridge University Press, Cambridge, 2006). Crossref, Google Scholar
3. M. Brion, Representations of quivers, in Geometric Methods in Representation Theory. I, Séminaires et Congrès, Volume 24 (Société Mathématique de France, Paris, 2012), pp. 103–144. Google Scholar
4. J. C. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, Advanced Courses in Mathematics - CRM Barcelona (Birkhäuser, Basel, 2002). Crossref, Google Scholar
5. F. U. Coelho and S.-X. Liu, Generalized path algebras, in Interactions Between Ring Theory and Representations of Algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 210, eds. F. van Oystaeyen and M. Saorin (Dekker, New York, 2000), pp. 53–66. Google Scholar
6. P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14(4) (2007) 589–596. Crossref, Web of Science, Google Scholar
7. R. Fröberg, Koszul algebras, in Advances in Commutative Ring Theory, Lecture Notes in Pure and Applied Mathematics, Vol. 205, eds. D. Dobbs, M. Fontana and S.-E. Kabbaj (Marcel Dekker, 1999), pp. 337–350. Google Scholar
8. K. R. Goodearl and R. B. Warfield Jr., An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, Vol. 61, 2nd edn. (Cambridge University Press, 2004). Crossref, Google Scholar
9. T. MathfrakHashi, Compact quantum groups of face type, Publ. Res. Inst. Math. Sci. 32(2) (1996) 351–369. Crossref, Web of Science, Google Scholar
10. H. Huang, C. Walton, E. Wicks and R. Won, Universal quantum semigroupoids, preprint (2020), arXiv:2008.00606. Google Scholar
11. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, Vol. 30 (American Mathematical Society, 2001). Crossref, Google Scholar
12. M. S. Molina, Algebras of quotients of path algebras, J. Algebra 319(12) (2008) 5265–5278. Crossref, Web of Science, Google Scholar
13. J. M. Moreno-Fernández and M. S. Molina, Graph algebras and the Gelfand-Kirillov dimension, J. Algebra Appl. 17(5) (2018) 1850095. Link, Web of Science, Google Scholar
14. H. Pfeiffer, Fusion categories in terms of graphs and relations, Quantum Topol. 2(4) (2011) 339–379. Crossref, Google Scholar
15. V. Ufnarovskiĭ, A growth criterion for graphs and algebras defined by words, Mat. Zametki 31(3) (1982) 465–472, 476. Google Scholar
16. C. Walton and X. Wang, On quantum groups associated to non-Noetherian regular algebras of dimension 2, Math. Z. 284(1–2) (2016) 543–574. Crossref, Web of Science, Google Scholar