# Yetter-Drinfel’d module

###### Definition 0.1.

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

 $\lambda_{M}(m)=\sum m_{(âˆ’1)}\otimes m_{0}$

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

 $\sum X^{1}m_{(âˆ’1)}\otimes(X^{2}.m_{(0)})_{(âˆ’1)}X^{3}\otimes(X^{2}.% m_{(0)})_{0}=\sum X^{1}(Y^{1}\times m)_{(âˆ’1)1}Y^{2}\otimes X^{2}\times(Y% ^{1}xm)_{(âˆ’1)2}\times Y^{3}\otimes X^{3}x(Y^{1}xm)_{(0)},$

and

 $\sum\epsilon(m_{(âˆ’1)})m_{0}=m,$

and

 $\sum h_{1}m_{(âˆ’1)}\otimes h_{2}\times m_{0}=\sum(h_{1}.m)_{(âˆ’1)}h_% {2}\otimes(h_{1}.m)_{0}.$

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.

## References

• 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. http://arxiv.org/PS_cache/math/pdf/0311/0311381v1.pdfMore Properties of Yetter-Drinfeld modules over Quasi-Hopf Algebras., Preprint.
 Title Yetter-Drinfel’d module Canonical name YetterDrinfeldModule Date of creation 2013-03-22 18:24:15 Last modified on 2013-03-22 18:24:15 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 19 Author bci1 (20947) Entry type Definition Classification msc 57T05 Classification msc 13-00 Classification msc 81R50 Classification msc 81R15 Classification msc 46L05 Classification msc 16W30 Synonym Drinfel’d module Synonym quasi-bialgebra Related topic QuantumGroups Related topic Module Related topic GrassmannHopfAlgebroidCategoriesAndGrassmannCategories Related topic GrassmanHopfAlgebrasAndTheirDualCoAlgebras Related topic C_cG Related topic LocallyCompact Related topic LocallyCompactGroupoids Related topic WeakHopfCAlgebra2 Related topic Bialgebra Related topic ExampleOfModule2 Defines H-module Defines bialgebras