
202 Algebraic Operads: An Algorithmic Companion
by the signs that come from compositions in S
−1
. Therefore, if we denote
the generator of As
q
and the generator of As
!
q
by the same symbol , the
pairing in question satisfies
*
,
+
= −
*
,
+
.
In addition, we of course have
*
,
+
= 0 =
*
,
+
.
The usual convention is to put
*
,
+
= 1,
*
,
+
= −1.
In this case, the dual of the defining relation
− q = 0
of As
q
is the relation
q − = 0.
In other words, As
!
q
∼
=
As
q
−1
(for q 6= 0).
Example 6.3.2.2. Consider the shuffle operad Lie
f
which, as we know from
Example 5.3.4.2, is the operad with just one defining relation
1 2
3
−
2 3
1
−
1 3
2
= 0.
In the shuffle case, in addition to the property highlighted in Example ...