B Homotopy Involutive Lie Bialgebras
89
B.3
IBL ∞ -Morphisms and Maurer–Cartan Elements
An L ∞ -morphism was defined by two equations (A.4). The first one involves the
comultiplication and implies that an L ∞ -morphism can be expressed by a linear
map from SA to A, cf. (A.5). We do not know of a suitable generalization of the
first equation in (A.4) to the case of I BL ∞ -algebras, but instead one can easily
generalize Eq. (A.5). The second equation of (A.4) is just saying that the morphism
commutes with the differentials and looks identically in the case of I BL ∞ -algebras.
Let (A , L ), (A , L ) be I BL ∞ -algebras. An I BL ∞ -morphism F ∈
Morph(A , A , x) is defined by
F =
∞
n=0
1
n!
f
∧n
◦ Δ n ,
F ◦ L
= L
◦ F
and
|F| = 0 ,
(B.11)
where
f =
∞
n=0
x
n−1 f
(n)
and
f
(n)
: SA
→ Σ
n A
.
The precise meaning of the morphism F can be found in Part II, Sect. 8.3, especially
Lemma 8.2. Recall that Σ n A = ⊕ n
i=1 A ∧i . We can therefore decompose f (n) into
a set of maps f n−g,g : SA → A ∧n−g , g ∈ {0, . . . , n − 1} in the same way as we
decomposed higher order coderivations. Expressed in terms of maps f n,g we have
f =
∞
n=1
∞
g=0
x
n+g−1 f
n,g .
(B.12)
Due to the lack of an algebraic relation governing the structure of an I BL ∞ -
morphism—an equation generalizing the first equation in (A.4)—it is not obvious
that the composition of two morphisms yields again a morphism. Nevertheless, this
can be shown to be true.
To complete this section, we finally state what a Maurer–Cartan element of an
I BL ∞ -algebra (A, L) is. Let c n,g ∈ A ∧n be of degree zero. The expression c =
∞
n=1
∞
g=0 x n+g−1 c n,g is called a Maurer–Cartan element of (A, L) if
L(e
c ) = 0 .
Again we can interpret a Maurer–Cartan element as a constant morphism from the
trivial I BL ∞ -algebra to (A, L). Here, the exponential is defined in the same way as
in the case of L ∞ -algebras, i.e., e c =
∞
n=0
1
n! c ∧n , now being a formal power series
with values in the completion of the symmetric algebra over A.
89
B.3
IBL ∞ -Morphisms and Maurer–Cartan Elements
An L ∞ -morphism was defined by two equations (A.4). The first one involves the
comultiplication and implies that an L ∞ -morphism can be expressed by a linear
map from SA to A, cf. (A.5). We do not know of a suitable generalization of the
first equation in (A.4) to the case of I BL ∞ -algebras, but instead one can easily
generalize Eq. (A.5). The second equation of (A.4) is just saying that the morphism
commutes with the differentials and looks identically in the case of I BL ∞ -algebras.
Let (A , L ), (A , L ) be I BL ∞ -algebras. An I BL ∞ -morphism F ∈
Morph(A , A , x) is defined by
F =
∞
n=0
1
n!
f
∧n
◦ Δ n ,
F ◦ L
= L
◦ F
and
|F| = 0 ,
(B.11)
where
f =
∞
n=0
x
n−1 f
(n)
and
f
(n)
: SA
→ Σ
n A
.
The precise meaning of the morphism F can be found in Part II, Sect. 8.3, especially
Lemma 8.2. Recall that Σ n A = ⊕ n
i=1 A ∧i . We can therefore decompose f (n) into
a set of maps f n−g,g : SA → A ∧n−g , g ∈ {0, . . . , n − 1} in the same way as we
decomposed higher order coderivations. Expressed in terms of maps f n,g we have
f =
∞
n=1
∞
g=0
x
n+g−1 f
n,g .
(B.12)
Due to the lack of an algebraic relation governing the structure of an I BL ∞ -
morphism—an equation generalizing the first equation in (A.4)—it is not obvious
that the composition of two morphisms yields again a morphism. Nevertheless, this
can be shown to be true.
To complete this section, we finally state what a Maurer–Cartan element of an
I BL ∞ -algebra (A, L) is. Let c n,g ∈ A ∧n be of degree zero. The expression c =
∞
n=1
∞
g=0 x n+g−1 c n,g is called a Maurer–Cartan element of (A, L) if
L(e
c ) = 0 .
Again we can interpret a Maurer–Cartan element as a constant morphism from the
trivial I BL ∞ -algebra to (A, L). Here, the exponential is defined in the same way as
in the case of L ∞ -algebras, i.e., e c =
∞
n=0
1
n! c ∧n , now being a formal power series
with values in the completion of the symmetric algebra over A.