Yetter-Drinfel’d module


Definition 0.1.

Let H be a quasi-bialgebra (http://planetmath.org/Bialgebra) with reassociator Φ. A left H-module M together with a left H-coaction λM:MHM,

λM(m)=m(−1)m0

is called a left Yetter-Drinfeld module if the following equalities hold, for all hH and mM:

X1m(−1)(X2.m(0))(−1)X3(X2.m(0))0=X1(Y1×m)(−1)1Y2X2×(Y1xm)(−1)2×Y3X3x(Y1xm)(0),

and

ϵ(m(−1))m0=m,

and

h1m(−1)h2×m0=(h1.m)(−1)h2(h1.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 fieldMathworldPlanetmath, 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