8.3 IBL ∞ -algebras
213
As proved in [12, Corollary 33], IBL ∞ -algebras with the above morphisms form
a category with the composition
f g := log
exp(f ) ◦ exp(g)
and the categorical unit
1 S(U ) = log
1 S(U )[[h]]
.
As argued in [12], IBL ∞ -algebras form a subcategory of the still bigger category of
Markl–Voronov algebras.
Example 8.5 Iterating the definition of the coproduct δ in S(U ), one obtains for the
exponential in (8.54) the expression
exp(f )(u 1 · · · · u n )
=
1
k!
)
a 1 ! · · · a k !
f (u σ (1) · · · · u σ (a 1 ) ) · · · f (u σ (n−a k +1) · · · · u σ (n) ),
where the summation runs over all permutations σ ∈ Σ n , all k ≥ 1, and all nonnegative integers a 1 , . . . , a k such that a 1 + · · · + a k = n. We recognize a formula
in [3, Section 5].
Example 8.6 The category of IBL ∞ -algebras contains a non-full subcategory
whose objects are L ∞ -algebras as in Example 8.3 and morphisms are k-linear
maps
f : S(U
) → S(U
)
such that
f (1) = 0, Δ
◦ exp(f ) = exp(f ) ◦ Δ
, and Im(f ) ⊂ U
.
Such a map automatically belongs to Lin
0
¯
h
S(U ), S(U )
. We leave as an exercise
to prove that
exp(f ) : S(U
) → S(U
)
is the unique extension of f into a morphism of unital algebras. We identify this
result as the dual to that of [7, Remark 5.3] describing the category of L ∞ -algebras
and their (weak) L ∞ -morphisms.