6.3 Operad Algebras
131
Another presentation of elements of W 0 , closer to the open string philosophy, is via
oriented spheres with one toothed hole:
whose teeth are indexed by S so that its cyclic order agrees with the one induced by
the orientation of the sphere.
6.3
Operad Algebras
The importance of operads is that they describe, via their representations, algebras
of specific types.
Definition 6.14 Let V be a graded vector space and s ∈ V ⊗ V a symmetric
degree 0 element. An algebra over a cyclic operad P, or a P-algebra, is a
morphism α : P → End (V ,s) .
Example 6.21 An algebra over the operad Com from Example 6.1 is the same as
a fully symmetric degree 0 linear map f : V ⊗3 → k such that the linear map
V ⊗4 → k
f
v 1 , v 2 , s
i
f
s
i , v 3 , v 4
(6.46)
is cyclically symmetric in v 1 , v 2 , v 3 , and v 4 . 8 Let us verify this statement.
A Com-algebra is a morphism α : Com → End (V ,s) . By Proposition 6.11, such
a morphism is determined by f := α(μ {1,2,3} ), where μ {1,2,3} satisfies (6.40). Such
an f is fully symmetric by the symmetry of μ {1,2,3} . Since α is an operad morphism,
α(μ {1,2,3} 3 ◦ 3 μ {3,4,5} ) = α(μ {1,2,3} ) 3 ◦ 3 α(μ {3,4,5} ),
while it is simple to identify, invoking the definition of the composition in the
endomorphism operad, the right-hand side term with expression (6.46). The cyclic
symmetry of (6.40) therefore implies the cyclic symmetry of (6.46). On the other
hand, it is easy to see that each f : V ⊗3 → k with the above properties determines
a unique morphism α : Com → End (V ,s) such that f := α(μ {1,2,3} ).
8 As before, we are using a variation on Sweedler’s convention s =
s
i ⊗s
i .
131
Another presentation of elements of W 0 , closer to the open string philosophy, is via
oriented spheres with one toothed hole:
whose teeth are indexed by S so that its cyclic order agrees with the one induced by
the orientation of the sphere.
6.3
Operad Algebras
The importance of operads is that they describe, via their representations, algebras
of specific types.
Definition 6.14 Let V be a graded vector space and s ∈ V ⊗ V a symmetric
degree 0 element. An algebra over a cyclic operad P, or a P-algebra, is a
morphism α : P → End (V ,s) .
Example 6.21 An algebra over the operad Com from Example 6.1 is the same as
a fully symmetric degree 0 linear map f : V ⊗3 → k such that the linear map
V ⊗4 → k
f
v 1 , v 2 , s
i
f
s
i , v 3 , v 4
(6.46)
is cyclically symmetric in v 1 , v 2 , v 3 , and v 4 . 8 Let us verify this statement.
A Com-algebra is a morphism α : Com → End (V ,s) . By Proposition 6.11, such
a morphism is determined by f := α(μ {1,2,3} ), where μ {1,2,3} satisfies (6.40). Such
an f is fully symmetric by the symmetry of μ {1,2,3} . Since α is an operad morphism,
α(μ {1,2,3} 3 ◦ 3 μ {3,4,5} ) = α(μ {1,2,3} ) 3 ◦ 3 α(μ {3,4,5} ),
while it is simple to identify, invoking the definition of the composition in the
endomorphism operad, the right-hand side term with expression (6.46). The cyclic
symmetry of (6.40) therefore implies the cyclic symmetry of (6.46). On the other
hand, it is easy to see that each f : V ⊗3 → k with the above properties determines
a unique morphism α : Com → End (V ,s) such that f := α(μ {1,2,3} ).
8 As before, we are using a variation on Sweedler’s convention s =
s
i ⊗s
i .