Twisted Associative Algebras and Shuffle Algebras 123
the partition ˜π
k
of I
k
, k = 1, 2, using the unique order preserving bijection
I
k
∼
=
{1, . . . , n
k
}. The shuffle (I
1
, I
2
)-product
µ
I
1
,I
2
(T
1
, T
2
) ∈ X
X
(n)
is the shuffle monomial (π, m), where π = ˜π
1
t ˜π
2
, m = m
1
m
2
.
These operations µ
I
1
,I
2
may be extended to unique bilinear operations
µ
I
1
,I
2
: T
X
(X)(n
1
) ⊗T
X
(X)(n
2
) → T
X
(X)(n).
Equipped with these operations, T
X
(X) is the free shuffle algebra gener-
ated by X. In addition to the notation T
X
(X), we will use the notation
T
X
(M), where M = {M(n)}
n≥0
is a collection of vector spaces for which
M(n) = span(X(n)) for all n ≥ 0.
Example 4.3.1.8. Let us consider the free shuffle algebra T
X
(1) from Exam-
ple 4.3.1.5. The four different shuffle products defined for the shuffle monomial ...