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Séminaire Lotharingien de Combinatoire, B62e (2010), 14 pp.

# Gérard H. E. Duchamp, Christophe Tollu, Karol A. Penson and Gleb A. Koshevoy

# Deformations of Algebras: Twisting and Perturbations

**Abstract.**
The framework used to prove the multiplicative deformation
of the algebra of Feynman-Bender diagrams is a *twisted shifted
dual law* (in fact, doubly twisted). We give here a clear interpretation of
its two parameters. The crossing parameter is a deformation of
the tensor structure, whereas the superposition parameter is a
perturbation of the shuffle coproduct which,
in its turn, can be interpreted as the diagonal restriction of a
superproduct. Here, we systematically explain these constructions
in detail.

Received: April 28, 2009.
Revised: September 6, 2010.
Accepted: November 26, 2010.
Final Version: November 30, 2010.

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