8.3 IBL ∞ -algebras
207
whenever g ≥ 1 are strongly homotopy Lie algebras [8], see the discussion of the
“tree level” in [9, page 372].
8.3
IBL ∞ -algebras
In this final section of the mathematical part we review, following closely [12],
a common generalization of loop homotopy algebras as well as Lie bialgebras. We
start by recalling some necessary auxiliary notions. Let A be a unital associative
commutative algebra and Δ : A → A a k-linear map. For n ≥ 0, consider the
iterated graded commutators
[[ . . . [Δ, L a 1 ], ...], L a n ] : A → A,
with L a denoting the operator of left multiplication by a ∈ A. By convention, we
just set the commutator of Δ with n = 0 left-multiplication operators to be Δ. We
call an operator Δ an order ≤ k differential operator if the iterated commutator with
any k + 1 left-multiplication operators vanish.
Now suppose that Δ(1) = 0. 8 Define
Φ
Δ
n (a 1 , . . . , a n ) := [[ . . . [Δ, L a 1 ], ...], L a n ](1) ∈ A.
In particular, Φ Δ
0 = 0. If Φ Δ
n = 0 for n > k, the operator Δ is called an order k
derivation [11, Section 1.2].
Example 8.1 Assume for simplicity that the degree of Δ is 0 and that A is ungraded,
the general graded case can be discussed analogously. For a, x ∈ A one has
[Δ, L a ](x) = Δ(ax) − aΔ(x). Invoking the unitality of A we immediately see
that Δ is an order 0 operator if and only if it is the left multiplication by Δ(1). Since
Φ Δ
1 = Δ, the only degree 0 derivations are trivial maps.
For a, b, x ∈ A one has
[Δ, L a ], L b
(x) = Δ(abx) − aΔ(bx) − Δ(ax)b +
aΔ(x)b. We leave as an exercise to show that Δ is an order ≤ 1 operator if and
only if it (uniquely) decomposes into the sum of the left multiplication L Δ(1) with a
derivation. Since
Φ
Δ
2 (a, b) = Δ(ab) − aΔ(b) − Δ(a)b,
the first order derivations are “ordinary” derivations, i.e. vector fields.
Definition 8.2 Let S(U ) be the polynomial algebra generated by a graded vector
space U and ¯
h a formal degree 0 symbol. An IBL ∞ -algebra structure on U [4],
[13, §4.2] is given by a degree 1, k[[ ¯
h]]-linear map Δ : S(U )[[ ¯
h]] → S(U )[[ ¯
h]]
8 All operators Δ in this section will share this property.
207
whenever g ≥ 1 are strongly homotopy Lie algebras [8], see the discussion of the
“tree level” in [9, page 372].
8.3
IBL ∞ -algebras
In this final section of the mathematical part we review, following closely [12],
a common generalization of loop homotopy algebras as well as Lie bialgebras. We
start by recalling some necessary auxiliary notions. Let A be a unital associative
commutative algebra and Δ : A → A a k-linear map. For n ≥ 0, consider the
iterated graded commutators
[[ . . . [Δ, L a 1 ], ...], L a n ] : A → A,
with L a denoting the operator of left multiplication by a ∈ A. By convention, we
just set the commutator of Δ with n = 0 left-multiplication operators to be Δ. We
call an operator Δ an order ≤ k differential operator if the iterated commutator with
any k + 1 left-multiplication operators vanish.
Now suppose that Δ(1) = 0. 8 Define
Φ
Δ
n (a 1 , . . . , a n ) := [[ . . . [Δ, L a 1 ], ...], L a n ](1) ∈ A.
In particular, Φ Δ
0 = 0. If Φ Δ
n = 0 for n > k, the operator Δ is called an order k
derivation [11, Section 1.2].
Example 8.1 Assume for simplicity that the degree of Δ is 0 and that A is ungraded,
the general graded case can be discussed analogously. For a, x ∈ A one has
[Δ, L a ](x) = Δ(ax) − aΔ(x). Invoking the unitality of A we immediately see
that Δ is an order 0 operator if and only if it is the left multiplication by Δ(1). Since
Φ Δ
1 = Δ, the only degree 0 derivations are trivial maps.
For a, b, x ∈ A one has
[Δ, L a ], L b
(x) = Δ(abx) − aΔ(bx) − Δ(ax)b +
aΔ(x)b. We leave as an exercise to show that Δ is an order ≤ 1 operator if and
only if it (uniquely) decomposes into the sum of the left multiplication L Δ(1) with a
derivation. Since
Φ
Δ
2 (a, b) = Δ(ab) − aΔ(b) − Δ(a)b,
the first order derivations are “ordinary” derivations, i.e. vector fields.
Definition 8.2 Let S(U ) be the polynomial algebra generated by a graded vector
space U and ¯
h a formal degree 0 symbol. An IBL ∞ -algebra structure on U [4],
[13, §4.2] is given by a degree 1, k[[ ¯
h]]-linear map Δ : S(U )[[ ¯
h]] → S(U )[[ ¯
h]]
8 All operators Δ in this section will share this property.