PlanetPhysics/Yetter Drinfeld Module

Let $H$ be a [[../Bialgebra/|quasi-bialgebra]] with reassociator $\mathrm{\Phi }$. A left $H$-module $M$ together with a left $H$-coaction ${\lambda }_M:M\to H\otimes M$, ${\lambda }_M(m)=\sum m_{(\hat{a}HR1)}\otimes m_0$ is called a left Yetter-Drinfeld [[../RModule/|module]] if the following equalities hold, for all $h\in H$ and $m\in M$:

$\sum X^1{m}_{(\hat{a}HR1)}\otimes (X^2.m_{(0)}{)}_{(\hat{a}HR1)}{X}^3\otimes (X^2.m_{(0)}{)}_0=\sum X^1(Y^1\times m{)}_{(\hat{a}HR1)1}{Y}^2\otimes X^2\times (Y^{1}xm{)}_{(\hat{a}HR1)2}\times Y^3\otimes X^{3}x(Y^{1}xm{)}_{(0)},$ and $\sum ϵ(m_{(\hat{a}HR1)})m_0=m,$ and

$\sum h_1{m}_{(\hat{a}HR1)}\otimes h_2\times m_0=\sum (h_1.m{)}_{(\hat{a}HR1)}{h}_2\otimes (h_1.m{)}_0.$

{\mathbf Remark} This module (ref.) is essential for solving the quasi--Yang--Baxter equation which is an important [[../Bijective/|relation]] in [[../PhysicalMathematics2/|mathematical physics]].

Drinfel'd modules

Let us consider a module that operates over a ring of [[../Bijective/|functions]] on a curve over a finite [[../CosmologicalConstant/|field]], which is called an elliptic module . Such modules were first studied by Vladimir Drinfel'd in 1973 and called accordingly Drinfel'd modules.