A
A ∞ - and L ∞ -Algebras
We review definitions of A ∞ - and L ∞ -algebras. In the following A =
n∈Z A n
will denote a graded vector space over some field k of characteristic 0 (more
generally we could consider a module A over some commutative ring R with
unit containing rational numbers). We will use the Koszul sign convention, that is,
we generate a sign (−1) xy whenever we permute two objects x and y with their
respective degrees denoted by the same symbols. If we permute several object, we
abbreviate the Koszul sign by (−1) . To simplify the exposition, we will assume in
this appendix all homogeneous pieces of the underlying graded vector spaces to be
finite-dimensional.
A.1
A ∞ -Algebras
Let us consider the tensor algebra of A
T A =
∞
n=0
A
⊗n ,
and the comultiplication Δ : T A → T A ⊗ T A defined by
Δ(a 1 ⊗ · · · ⊗ a n ) =
n
i=0
(a 1 ⊗ · · · ⊗ a i ) ⊗ (a i+1 ⊗ · · · ⊗ a n ) .
The comultiplication Δ makes T A a coassociative coalgebra, i.e.,
(Δ ⊗ 1) ◦ Δ = (1 ⊗ Δ) ◦ Δ .
© 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
77