
138 Algebraic Operads: An Algorithmic Companion
Definition 4.6.3.2 (Commutative shuffle algebra). A shuffle algebra A for
which
µ
I
1
,I
2
(a
1
, a
2
) = µ
I
2
,I
1
(a
2
, a
1
)
whenever {1, . . . , n} = I
1
t I
2
, a
1
∈ A(|I
1
|), a
2
∈ A(|I
2
|), is said to be com-
mutative.
The most important example of a twisted commutative algebra is the ten-
sor algebra of a vector space.
Proposition 4.6.3.3. The tensor algebra T (V ) with its shuffle algebra struc-
ture is commutative. In fact, it is free as a commutative shuffle algebra.
Proof. The first statement is trivially true; it is essentially explained in the
introduction to this chapter. The second statement is left as an exercise for
the