Abstract
A dynamical Yang–Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang–Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang–Baxter maps. Our key observation is that the category of dynamical sets over a set Λ introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang–Baxter maps, can be embedded into the category of quivers with vertices Λ. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang–Baxter maps and extend his construction to obtain a new class of dynamical Yang–Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang–Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.
| Original language | English |
|---|---|
| Pages (from-to) | 47-80 |
| Number of pages | 34 |
| Journal | Journal of Algebra |
| Volume | 507 |
| DOIs | |
| Publication status | Published - 2018 Aug 1 |
Keywords
- Braided object
- Braided quiver
- Dynamical Yang–Baxter map
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
Quiver-theoretical approach to dynamical Yang–Baxter maps. / Matsumoto, Diogo Kendy; Shimizu, Kenichi.
In: Journal of Algebra, Vol. 507, 01.08.2018, p. 47-80.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Quiver-theoretical approach to dynamical Yang–Baxter maps
AU - Matsumoto, Diogo Kendy
AU - Shimizu, Kenichi
PY - 2018/8/1
Y1 - 2018/8/1
N2 - A dynamical Yang–Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang–Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang–Baxter maps. Our key observation is that the category of dynamical sets over a set Λ introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang–Baxter maps, can be embedded into the category of quivers with vertices Λ. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang–Baxter maps and extend his construction to obtain a new class of dynamical Yang–Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang–Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.
AB - A dynamical Yang–Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang–Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang–Baxter maps. Our key observation is that the category of dynamical sets over a set Λ introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang–Baxter maps, can be embedded into the category of quivers with vertices Λ. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang–Baxter maps and extend his construction to obtain a new class of dynamical Yang–Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang–Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.
KW - Braided object
KW - Braided quiver
KW - Dynamical Yang–Baxter map
UR - http://www.scopus.com/inward/record.url?scp=85045243126&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85045243126&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2018.04.003
DO - 10.1016/j.jalgebra.2018.04.003
M3 - Article
AN - SCOPUS:85045243126
VL - 507
SP - 47
EP - 80
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
ER -