
Chapter 4
Twisted Associative Algebras and
Shuffle Algebras
The goal of this chapter is to discuss the algebraic structures that are some-
where in between associative algebras and operads: twisted associative alge-
bras and shuffle algebras. Even though these can in principle be treated in
the more general operadic context, this topic is much easier to digest indepen-
dently, and it serves as a gentle introduction to the philosophy behind shuffle
operads that we will discuss later.
4.1 Introduction
When working with the tensor algebra of a vector space V , we use the
concatenation of tensors to define products of elements. More precisely, for
two decomposable ...