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.[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.

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