## Infinite Analysis Seminar Tokyo

Date, time & place Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.)

### 2015/01/22

13:00-16:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Diogo Kendy Matsumoto (Faculty of Science and Engineerring, Waseda University) 13:00-14:30
A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)
[ Abstract ]
Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.
Youichi Shibukawa (Department of Mathematics, Hokkaido University) 15:00-16:30
Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)
[ Abstract ]
A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.