Yetter-Drinfel'd module

Definition 0.1   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$ :

$\displaystyle \sum X^1 m_{(\widehat{a} H R1)} \otimes (X^2 . m_{(0)})_{(\wideha...
...\times (Y^1 x m)_{(\widehat{a}H R1)2} \times Y^3 \otimes X^3 x (Y^1 x m)_{(0)},$

and $ \sum \epsilon(m_{(\widehat{a} H R1)})m_0 = m ,$ and

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

Remark This module (ref.[1]) 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.

Bibliography

1
Bulacu, D, Caenepeel, S, Torrecillas, B, Doi-Hopf modules and Yetter-Drinfeld modules for quasi-Hopf algebras. Communications in Algebra, 34 (9), pp. 3413-3449, 2006.

2
D. Bulacu, S. Caenepeel, A and F. Panaite. 2003. More Properties of Yetter-Drinfeld modules over Quasi-Hopf Algebras., Preprint.



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