208
8 Structures Relevant to Physics
satisfying Δ 2 = 0, Δ(1) = 0, which moreover decomposes into a sum
Δ = Δ 1 + ¯
hΔ 2 + ¯
h
2 Δ 3 + · · · ,
(8.43)
where
Δ k : S(U )[[ ¯
h]] → S(U )[[ ¯
h]]
(8.44)
is an order ≤ k differential operator on the polynomial algebra S(U )[[ ¯
h]].
As proved in [9, Proposition 3], an order ≤ k differential operator as in (8.44)
with Δ(1) = 0 is determined by its restrictions
Δ k | S t (U )[[¯ h]] : S
t (U )[[ ¯
h]] → S(U )[[ ¯
h]]
to the subspaces S t (U ) ⊂ S(U ) of polynomials of length t with 1 ≤ t ≤ k.
These restrictions are, due to the assumed k[[ ¯
h]]-linearity, in turn determined by
their restrictions
ω k,t := Δ k | S t (U ) : S
t (U ) → S(U )[[ ¯
h]]
(8.45)
and thus, after singling out the coefficients at ¯
h g , by the family
ω k,t,g := Δ k | S t (U ) : S
t (U ) → S(U )
such that ω k,t =
g≥0 ω k,t,g ¯
h
g . Moreover, each ω k,t,g is determined by a family
ω
s
k,t,g : S
t (U ) → S
s (U ), s ≥ 0,
(8.46)
which is locally finite in the sense that, for a given u ∈ S t (U ), ω
s
k,t,g (u) = 0 for
only finitely many s.
On the other hand, backtracking the above procedure we readily see that each
family of locally finite degree +1 maps
ω
s
k,t,g : S
t (U ) → S
s (U ) | k ≥ 1, 1 ≤ t ≤ k, g, s ≥ 0
(8.47)
as in (8.46) assembles into a map Δ in (8.43). The one-to-one correspondence
{ω
s
k,t,g } ↔ Δ can be made explicit using the calculations in Section 3 of [9]. General
formulas are however clumsy so we do not include them here.
Remark 8.1 We saw that IBL ∞ -algebras can be viewed as structures with infinitely
many degree +1 operations (8.46) satisfying an infinite set of axioms obtained
by assembling them into Δ and requiring Δ 2 = 0. Since IBL ∞ -algebras possess
structure operations with several inputs and several outputs, they are not algebras
over operads, but over more general objects called PROPs recalled e.g. in [10].
8 Structures Relevant to Physics
satisfying Δ 2 = 0, Δ(1) = 0, which moreover decomposes into a sum
Δ = Δ 1 + ¯
hΔ 2 + ¯
h
2 Δ 3 + · · · ,
(8.43)
where
Δ k : S(U )[[ ¯
h]] → S(U )[[ ¯
h]]
(8.44)
is an order ≤ k differential operator on the polynomial algebra S(U )[[ ¯
h]].
As proved in [9, Proposition 3], an order ≤ k differential operator as in (8.44)
with Δ(1) = 0 is determined by its restrictions
Δ k | S t (U )[[¯ h]] : S
t (U )[[ ¯
h]] → S(U )[[ ¯
h]]
to the subspaces S t (U ) ⊂ S(U ) of polynomials of length t with 1 ≤ t ≤ k.
These restrictions are, due to the assumed k[[ ¯
h]]-linearity, in turn determined by
their restrictions
ω k,t := Δ k | S t (U ) : S
t (U ) → S(U )[[ ¯
h]]
(8.45)
and thus, after singling out the coefficients at ¯
h g , by the family
ω k,t,g := Δ k | S t (U ) : S
t (U ) → S(U )
such that ω k,t =
g≥0 ω k,t,g ¯
h
g . Moreover, each ω k,t,g is determined by a family
ω
s
k,t,g : S
t (U ) → S
s (U ), s ≥ 0,
(8.46)
which is locally finite in the sense that, for a given u ∈ S t (U ), ω
s
k,t,g (u) = 0 for
only finitely many s.
On the other hand, backtracking the above procedure we readily see that each
family of locally finite degree +1 maps
ω
s
k,t,g : S
t (U ) → S
s (U ) | k ≥ 1, 1 ≤ t ≤ k, g, s ≥ 0
(8.47)
as in (8.46) assembles into a map Δ in (8.43). The one-to-one correspondence
{ω
s
k,t,g } ↔ Δ can be made explicit using the calculations in Section 3 of [9]. General
formulas are however clumsy so we do not include them here.
Remark 8.1 We saw that IBL ∞ -algebras can be viewed as structures with infinitely
many degree +1 operations (8.46) satisfying an infinite set of axioms obtained
by assembling them into Δ and requiring Δ 2 = 0. Since IBL ∞ -algebras possess
structure operations with several inputs and several outputs, they are not algebras
over operads, but over more general objects called PROPs recalled e.g. in [10].