Revista de la
Unión Matemática Argentina
Two constructions of bialgebroids and their relations
Yudai Otsuto

Volume 67, no. 1 (2024), pp. 339–395    

Published online: June 25, 2024

https://doi.org/10.33044/revuma.2296

Download PDF

Abstract

We generalize the construction of face algebras by Hayashi and obtain a left bialgebroid $\mathfrak{A}(w)$. There are some relations between the left bialgebroid $\mathfrak{A}(w)$ and the generalized Shibukawa–Takeuchi left bialgebroid $A_{\sigma}$.

References

  1. R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193–228.  DOI  MR  Zbl
  2. S. Bennoun and H. Pfeiffer, Weak bialgebras of fractions, J. Algebra 385 (2013), 145–163.  DOI  MR  Zbl
  3. G. Böhm, F. Nill, and K. Szlachányi, Weak Hopf algebras. I. Integral theory and $C^*$-structure, J. Algebra 221 no. 2 (1999), 385–438.  DOI  MR  Zbl
  4. G. Böhm and K. Szlachányi, Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra 274 no. 2 (2004), 708–750.  DOI  MR  Zbl
  5. V. G. Drinfelʹd, On some unsolved problems in quantum group theory, in Quantum groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, Berlin, 1992, pp. 1–8.  DOI  MR  Zbl
  6. P. Etingof, T. Schedler, and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 no. 2 (1999), 169–209.  DOI  MR  Zbl
  7. P. Etingof and A. Varchenko, Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys. 196 no. 3 (1998), 591–640.  DOI  MR  Zbl
  8. L. D. Faddeev, N. Y. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, in Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 129–139.  DOI  MR  Zbl
  9. J.-L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville string field theory, Nuclear Phys. B 238 no. 1 (1984), 125–141.  DOI  MR
  10. T. Hayashi, Quantum group symmetry of partition functions of IRF models and its application to Jones' index theory, Comm. Math. Phys. 157 no. 2 (1993), 331–345.  MR  Zbl Available at http://projecteuclid.org/euclid.cmp/1104253942.
  11. T. Hayashi, Quantum groups and quantum semigroups, J. Algebra 204 no. 1 (1998), 225–254.  DOI  MR  Zbl
  12. D. K. Matsumoto and K. Shimizu, Quiver-theoretical approach to dynamical Yang-Baxter maps, J. Algebra 507 (2018), 47–80.  DOI  MR  Zbl
  13. J. B. McGuire, Study of exactly soluble one-dimensional $N$-body problems, J. Math. Phys. 5 (1964), 622–636.  DOI  MR  Zbl
  14. F. Nill, Axioms for weak bialgebras, 1998. arXiv:math/9805104 [math.QA].
  15. Y. Otsuto, Two constructions of Hopf algebroids based on the FRT construction and their relations, Ph.D. thesis, Hokkaido University, 2022.  DOI
  16. Y. Otsuto and Y. Shibukawa, FRT construction of Hopf algebroids, Toyama Math. J. 42 (2021), 51–72.  MR  Zbl
  17. P. Schauenburg, Duals and doubles of quantum groupoids ($\times_R$-Hopf algebras), in New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 273–299.  DOI  MR  Zbl
  18. P. Schauenburg, Weak Hopf algebras and quantum groupoids, in Noncommutative geometry and quantum groups (Warsaw, 2001), Banach Center Publ. 61, Polish Acad. Sci. Inst. Math., Warsaw, 2003, pp. 171–188.  DOI  MR  Zbl
  19. Y. Shibukawa, Dynamical Yang-Baxter maps, Int. Math. Res. Not. no. 36 (2005), 2199–2221.  DOI  MR  Zbl
  20. Y. Shibukawa, Dynamical Yang-Baxter maps with an invariance condition, Publ. Res. Inst. Math. Sci. 43 no. 4 (2007), 1157–1182.  DOI  MR  Zbl
  21. Y. Shibukawa, Hopf algebroids and rigid tensor categories associated with dynamical Yang-Baxter maps, J. Algebra 449 (2016), 408–445.  DOI  MR  Zbl
  22. Y. Shibukawa and M. Takeuchi, FRT construction for dynamical Yang-Baxter maps, J. Algebra 323 no. 6 (2010), 1698–1728.  DOI  MR  Zbl
  23. M. Takeuchi, Groups of algebras over $A\otimes \overline A$, J. Math. Soc. Japan 29 no. 3 (1977), 459–492.  DOI  MR  Zbl
  24. A. Veselov, Yang-Baxter maps: dynamical point of view, in Combinatorial aspect of integrable systems, MSJ Mem. 17, Math. Soc. Japan, Tokyo, 2007, pp. 145–167.  DOI  MR  Zbl
  25. C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312–1315.  DOI  MR  Zbl