B
Homotopy Involutive Lie Bialgebras
Homotopy algebras as reviewed in Appendix A are suitable for describing the
algebraic structures of classical open-closed string field theory. If one tries to
describe quantum open-closed string field theory—with the set of vertices satisfying
the full quantum BV master equation—in the framework of homotopy algebras,
the appropriate language is that of homotopy involutive Lie bialgebras, or I BL ∞ -
algebras. 1 An I BL ∞ -algebra is a generalization of an L ∞ -algebra. Its axioms are
formulated in terms of higher order coderivations—a concept that will be introduced
in the next section—and requires an auxiliary parameter x ∈ k (later on we will
identify that parameter with ¯
h). We will also recall the notion of morphisms and
Maurer–Cartan elements in the context of I BL ∞ -algebras. Our exposition is based
on work of Cieliebak, Fukaya, and Latschev cited at the end of Chap. 5. In the
following, we collect their results (in a slightly different notation) to make our
exposition self-contained. An alternative description is provided by Sect. 8.3.
B.1
Higher Order Coderivations
We already know what a coderivation (of order one) on SA is, cf. Eq. (A.3). We
defined it by an algebraic equation involving the comultiplication Δ. The essence of
that equation was that a coderivation D ∈ Coder(SA) was uniquely determined by
a homomorphism d ∈ Hom(SA, A). Explicitly we had
D ◦ i n =
i+j =n
σ
(d i ∧ 1
∧j ) ◦ σ ,
( B . 1 )
where π 1 ◦ D = d.
1 As we already mentioned, an alternative description using the language of quantum open-closed
homotopy algebra is also possible.
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3
85
Homotopy Involutive Lie Bialgebras
Homotopy algebras as reviewed in Appendix A are suitable for describing the
algebraic structures of classical open-closed string field theory. If one tries to
describe quantum open-closed string field theory—with the set of vertices satisfying
the full quantum BV master equation—in the framework of homotopy algebras,
the appropriate language is that of homotopy involutive Lie bialgebras, or I BL ∞ -
algebras. 1 An I BL ∞ -algebra is a generalization of an L ∞ -algebra. Its axioms are
formulated in terms of higher order coderivations—a concept that will be introduced
in the next section—and requires an auxiliary parameter x ∈ k (later on we will
identify that parameter with ¯
h). We will also recall the notion of morphisms and
Maurer–Cartan elements in the context of I BL ∞ -algebras. Our exposition is based
on work of Cieliebak, Fukaya, and Latschev cited at the end of Chap. 5. In the
following, we collect their results (in a slightly different notation) to make our
exposition self-contained. An alternative description is provided by Sect. 8.3.
B.1
Higher Order Coderivations
We already know what a coderivation (of order one) on SA is, cf. Eq. (A.3). We
defined it by an algebraic equation involving the comultiplication Δ. The essence of
that equation was that a coderivation D ∈ Coder(SA) was uniquely determined by
a homomorphism d ∈ Hom(SA, A). Explicitly we had
D ◦ i n =
i+j =n
σ
(d i ∧ 1
∧j ) ◦ σ ,
( B . 1 )
where π 1 ◦ D = d.
1 As we already mentioned, an alternative description using the language of quantum open-closed
homotopy algebra is also possible.
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3
85