8.3 IBL ∞ -algebras
211
structure equation Δ 2 = 0 for Δ assembling the operations in (8.52) is equivalent
to the axioms of a loop homotopy algebra with the operations δ
g
n acting on the
differential graded vector space (V , d) equipped with the closed symmetric degree
+1 element s ∈ V ⊗ V .
The basic difference between the description of loop homotopy algebras given
in Sect. 8.2 and the one in Example 8.4 above is that there the bilinear form B =
ω 0
2,2,0 is considered as a structure operation, while in Sect. 8.2 the corresponding
symmetric element was fixed from the beginning.
Morphism Besides the commutative associative multiplication, the polynomial
algebra S(U ) bears also the coproduct δ : S(U ) → S(U )⊗S(U ) which turns it into
a commutative associative cocommutative coassociative coalgebra. The coproduct
given by the formula
δ(u 1 · · · · u n ) =
(σ )
a! b!
[u σ (1) · · · · u σ (a) ] ⊗ · · · ⊗ [u σ (a+1) · · · · u σ (n) ],
where the summation runs over all permutations σ ∈ Σ k and integers a, b ≥ 0 such
that a + b = n. As always, (σ ) denotes the Koszul sign and by we denote the
standard commutative product of S(U ).
It is well known, see e.g. [5, §III.3], that the space Lin k (A, C) of linear maps
from a commutative associative algebra A with multiplication μ to a cocommutative coassociative coalgebra C with comultiplication δ admits the commutative
coassociative convolution product defined as
f f g := μ(f ⊗ g)δ, f, g ∈ Lin k (A, C).
In particular, Lin k
S(U ), S(U )
with the convolution product is a commutative
associative algebra. One easily sees that the composition
e := S(U
) S
0 (U
) ∼ = k ∼ = S
0 (U
) )→ S(U
) ∈ Lin k
S(U
), S(U
)
of the natural projection followed by the natural inclusion is the unit for . The
above constructions extend by the k[[ ¯
h]]-linearity to the space
Lin k[[¯ h]]
S(U
)[[ ¯
h]], S(U
)[[ ¯
h]]
of k[[ ¯
h]]-linear maps which we will, for brevity, denote by Lin ¯
h
S(U ), S(U )
believing that the reader will not be too confused by this shorthand.
We denote finally by Lin
0
¯
h
S(U ), S(U )
the subset of Lin ¯
h
S(U ), S(U )
consisting of k[[ ¯
h]]-linear maps such that
f (1) ∈ S(U
)[[ ¯
h]].
(8.53)
211
structure equation Δ 2 = 0 for Δ assembling the operations in (8.52) is equivalent
to the axioms of a loop homotopy algebra with the operations δ
g
n acting on the
differential graded vector space (V , d) equipped with the closed symmetric degree
+1 element s ∈ V ⊗ V .
The basic difference between the description of loop homotopy algebras given
in Sect. 8.2 and the one in Example 8.4 above is that there the bilinear form B =
ω 0
2,2,0 is considered as a structure operation, while in Sect. 8.2 the corresponding
symmetric element was fixed from the beginning.
Morphism Besides the commutative associative multiplication, the polynomial
algebra S(U ) bears also the coproduct δ : S(U ) → S(U )⊗S(U ) which turns it into
a commutative associative cocommutative coassociative coalgebra. The coproduct
given by the formula
δ(u 1 · · · · u n ) =
(σ )
a! b!
[u σ (1) · · · · u σ (a) ] ⊗ · · · ⊗ [u σ (a+1) · · · · u σ (n) ],
where the summation runs over all permutations σ ∈ Σ k and integers a, b ≥ 0 such
that a + b = n. As always, (σ ) denotes the Koszul sign and by we denote the
standard commutative product of S(U ).
It is well known, see e.g. [5, §III.3], that the space Lin k (A, C) of linear maps
from a commutative associative algebra A with multiplication μ to a cocommutative coassociative coalgebra C with comultiplication δ admits the commutative
coassociative convolution product defined as
f f g := μ(f ⊗ g)δ, f, g ∈ Lin k (A, C).
In particular, Lin k
S(U ), S(U )
with the convolution product is a commutative
associative algebra. One easily sees that the composition
e := S(U
) S
0 (U
) ∼ = k ∼ = S
0 (U
) )→ S(U
) ∈ Lin k
S(U
), S(U
)
of the natural projection followed by the natural inclusion is the unit for . The
above constructions extend by the k[[ ¯
h]]-linearity to the space
Lin k[[¯ h]]
S(U
)[[ ¯
h]], S(U
)[[ ¯
h]]
of k[[ ¯
h]]-linear maps which we will, for brevity, denote by Lin ¯
h
S(U ), S(U )
believing that the reader will not be too confused by this shorthand.
We denote finally by Lin
0
¯
h
S(U ), S(U )
the subset of Lin ¯
h
S(U ), S(U )
consisting of k[[ ¯
h]]-linear maps such that
f (1) ∈ S(U
)[[ ¯
h]].
(8.53)