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Yetter-Drinfel'd module (Definition)
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.



"Yetter-Drinfel'd module" is owned by bci1.

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Other names:  Yetter module
Also defines:  Drinfel'd module, coaction, quasi-bialgebra, quasi-Hopf algebras, left $H$-coaction, left Yetter-Drinfel'd module, elliptic module, quasi--Yang--Baxter equation
Keywords:  Drinfel'd module, coaction, quasi-Hopf algebras, quasi-bialgebra, left $H$-coaction, left Yetter-Drinfel'd module, elliptic module, quasi--Yang--Baxter equation

Cross-references: field, functions, mathematical physics, relation, module

This is version 7 of Yetter-Drinfel'd module, born on 2009-04-24, modified 2009-04-24.
Object id is 689, canonical name is YetterDrinfeldModule.
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Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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