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6 Operads
One can also give the following more standard incarnation of Com-algebras.
Define a degree 0 bilinear operation μ : V ⊗ V → V by
μ(a, b) :=
s
i f (s
i , a, b), a, b ∈ V .
If s is non-degenerate and s −1 = B, then the assignment f ↔ μ defines a
one-to-one correspondence between Com-algebra structures f and commutative
associative multiplications μ on V such that
B
μ(a, b), c
= B
a, μ(b, c)
.
These structures are commutative non-unital versions of Frobenius algebras.
Let us denote by End (V ,s) the endomorphism operad End (V ,s) considered as a
non-Σ cyclic operad, i.e.,
End (V ,s) := (V ,s) ),
where is the forgetful functor of (6.41). By the standard properties of adjunctions,
for any non-Σ cyclic operad P there exists a one-to-one correspondence between
morphism
α : Sym(P) → End (V ,s)
in the category of cyclic operads, and morphism
α : P → End (V ,s)
(6.47)
in the category of non-Σ cyclic operads. Therefore an algebra over the symmetrization Sym(P) is the same as a morphism (6.47). The description of Ass =
Sym(Ass)-algebras in the following example thus easily follows from Proposition 6.8.
Example 6.22 An algebra over the operad Ass from Example 6.18 is the same as
a degree 0 cyclically symmetric linear map f : V ⊗3 → k such that the linear map
V ⊗4 → k defined by
f (v 1 , v 2 , s
i )f (s
i , v 3 , v 4 )
is cyclically symmetric, too.
Define as in Example 6.21 μ : V ⊗ V → V by μ(a, b) :=
s
i f (s
i , a, b). If s
is non-degenerate and s −1 = B, then the assignment f → μ defines an one-to-one
correspondence between Ass-algebra structures f and associative, not necessarily