PlanetPhysics/Yetter Drinfeld Module

Let $$H$$ be a quasi-bialgebra with reassociator $$\Phi$$. A left $$H$$-module $$M$$ together with a left $$H$$-coaction $$\lambda_M : M \to H \otimes M$$, $$\lambda_M (m) = \sum m_{(\widehat{a} H R1)} \otimes m_0$$ is called a left Yetter-Drinfeld module if the following equalities hold, for all $$h \in H$$ and $$m \in M$$:

$$\sum X^1 m_{(\widehat{a} H R1)} \otimes (X^2 . m_{(0)})_{(\widehat{a} H R1)} X^3 \otimes (X^2 . m_{(0)})_0 = \sum X^1(Y^1 \times m)_{(\widehat{a} H R1)1} Y^2 \otimes X^2 \times (Y^1 x m)_{(\widehat{a}H R1)2} \times Y^3 \otimes X^3 x (Y^1 x m)_{(0)},$$ and $$ \sum \epsilon(m_{(\widehat{a} H R1)})m_0 = m ,$$ and

$$ \sum h_1 m_{(\widehat{a}H R1)} \otimes h_2 \times m_0 = \sum (h_1 . m)_{(\widehat{a} H R1)} h_2 \otimes (h_1 . m)_0.$$

{\mathbf Remark} This module (ref. ) is essential for solving the quasi--Yang--Baxter equation which is an important 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 accordingly Drinfel'd modules.