Two constructions of bialgebroids and their relations
DOI:
https://doi.org/10.33044/revuma.2296Abstract
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 $A_{\sigma}$.
Downloads
References
R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193–228. DOI MR Zbl
S. Bennoun and H. Pfeiffer, Weak bialgebras of fractions, J. Algebra 385 (2013), 145–163. DOI MR Zbl
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
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
V. G. Drinfel'd, 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
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
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
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
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
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.
T. Hayashi, Quantum groups and quantum semigroups, J. Algebra 204 no. 1 (1998), 225–254. DOI MR Zbl
D. K. Matsumoto and K. Shimizu, Quiver-theoretical approach to dynamical Yang-Baxter maps, J. Algebra 507 (2018), 47–80. DOI MR Zbl
J. B. McGuire, Study of exactly soluble one-dimensional $N$-body problems, J. Math. Phys. 5 (1964), 622–636. DOI MR Zbl
F. Nill, Axioms for weak bialgebras, 1998. arXiv:math/9805104 [math.QA].
Y. Otsuto, Two constructions of Hopf algebroids based on the FRT construction and their relations, Ph.D. thesis, Hokkaido University, 2022. DOI
Y. Otsuto and Y. Shibukawa, FRT construction of Hopf algebroids, Toyama Math. J. 42 (2021), 51–72. MR Zbl
P. Schauenburg, Duals and doubles of quantum groupoids ($×_R$-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
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
Y. Shibukawa, Dynamical Yang-Baxter maps, Int. Math. Res. Not. no. 36 (2005), 2199–2221. DOI MR Zbl
Y. Shibukawa, Dynamical Yang-Baxter maps with an invariance condition, Publ. Res. Inst. Math. Sci. 43 no. 4 (2007), 1157–1182. DOI MR Zbl
Y. Shibukawa, Hopf algebroids and rigid tensor categories associated with dynamical Yang-Baxter maps, J. Algebra 449 (2016), 408–445. DOI MR Zbl
Y. Shibukawa and M. Takeuchi, FRT construction for dynamical Yang-Baxter maps, J. Algebra 323 no. 6 (2010), 1698–1728. DOI MR Zbl
M. Takeuchi, Groups of algebras over A⊗Ā, J. Math. Soc. Japan 29 no. 3 (1977), 459–492. DOI MR Zbl
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
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
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Yudai Otsuto
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.