Definition 0.1. Let H be a quasi-bialgebra with reassociator Φ. A left H-module M
together with a left H-coaction λM : M → H ⊗ M, λM(m) = ∑
m(aHR1) ⊗ m0 is called a
left Yetter-Drinfeld module if the following equalities hold, for all h ∈ H and m ∈ M:
and ∑
𝜖(m(aHR1))m0 = m, and
Remark This module (ref.[1]) is essential for solving the quasi–Yang–Baxter equation which is an
important relation in mathematical physics.
0.1 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. More Properties of Yetter-Drinfeld
modules over Quasi-Hopf Algebras., Preprint.
