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Yetter-Drinfel'd module
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(Definition)
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Definition 0.1 Let $H$ be a quasi-bialgebra with reassociator $\Phi$ . A left $H$ -module $M$ together with a left $H$ -coaction $\lambda_M : M \to H \otimes M,$ $$\lambda_M (m) = \sum m_{(−1)} \otimes m_0$$ is called a left Yetter-Drinfeld module if the following equalities hold, for all $h \in H$ and $m \in M :$
$$\sum X^1 m_{(−1)} \otimes (X^2 . m_{(0)})_{(−1)} X^3 \otimes (X^2 . m_{(0)})_0 = \sum X^1(Y^1 \times m)_{(−1)1} Y^2 \otimes X^2 \times (Y^1 x m)_{(−1)2} \times Y^3 \otimes X^3 x (Y^1 x m)_{(0)},$$ and
$$ \sum \epsilon(m_{(−1)})m_0 = m ,$$ and
$$ \sum h_1 m_{(−1)} \otimes h_2 \times m_0 = \sum (h_1 . m)_{(−1)} h_2 \otimes (h_1 . m)_0.$$
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.
- 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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"Yetter-Drinfel'd module" is owned by bci1.
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See Also: quantum groups, module, Grassmann-Hopf algebroid categories and Grassmann categories, Grassmann-Hopf algebras and coalgebras\gebras, locally compact, locally compact groupoids, weak Hopf C*-algebra, bialgebra, examples of modules
| Other names: |
Drinfel'd module, quasi-bialgebra |
| Also defines: |
H-module, bialgebras |
| Keywords: |
quantum double, Yetter-Drinfel'd module, quasi-Yang-Baxter equation, quasi-Hopf algebra, Drinfeld module, elliptic module |
This object's parent.
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Cross-references: finite field, curve, functions, ring, relation, equation, equalities, module
There are 10 references to this entry.
This is version 16 of Yetter-Drinfel'd module, born on 2008-09-20, modified 2008-10-23.
Object id is 11051, canonical name is QuantumDouble.
Accessed 1811 times total.
Classification:
| AMS MSC: | 16W30 (Associative rings and algebras :: Rings and algebras with additional structure :: Coalgebras, bialgebras, Hopf algebras ; rings, modules, etc. on which these act) | | | 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras) | | | 81R15 (Quantum theory :: Groups and algebras in quantum theory :: Operator algebra methods) | | | 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods) | | | 13-00 (Commutative rings and algebras :: General reference works ) | | | 57T05 (Manifolds and cell complexes :: Homology and homotopy of topological groups and related structures :: Hopf algebras) |
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Pending Errata and Addenda
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