Yetter-Drinfel’d module

Definition 0.1.

Let $H$ be a quasi-bialgebra (http://planetmath.org/Bialgebra) with reassociator $\mathrm{\Phi }$. A left $H$-module$M$ together with a left $H$-coaction ${\lambda }_M:M\to H\otimes M,$

$${\lambda }_M(m)=\sum m_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)}\otimes m_0$$

is called a left Yetter-Drinfeld module if the following equalities hold, for all$h\in H$ and $m\in M:$

$$\sum X^1{m}_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)}\otimes {(X^2.m_{(0)})}_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)}{X}^3\otimes {(X^2.m_{(0)})}_0=\sum X^1{(Y^1\times m)}_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)1}{Y}^2\otimes X^2\times {(Y^{1}xm)}_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)2}\times Y^3\otimes X^{3}x{(Y^{1}xm)}_{(0)},$$

and

$$\sum ϵ(m_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)})m_0=m,$$

and

$$\sum h_1{m}_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)}\otimes h_2\times m_0=\sum {(h_1.m)}_{(Ã\mathrm{¢}Ë\mathrm{†}â\mathrm{€}\mathrm{™}1)}{h}_2\otimes {(h_1.m)}_0.$$

Remark:This module (ref.[1]) is essential for solving the quasi-Yang-Baxter equation which isan important relation in Mathematical Physics.
Drinfel’d modules:Let us consider a module that operates over a ring of functions on a curve over a finite field, which is called an elliptic module. Such modules were first studied by Vladimir Drinfel’d in 1973 and called accordinglyDrinfel’d modules.