Symmetric Monoidal Categories for Operads - Index of

4.3 Enveloping Algebras 89
Proof. The construction of proposition 4.1.11 implies that the enveloping
algebra of A is defined by a reflexive coequalizer of the form:
P[S(P,A)](1)
φs 0
φd 0
φd 1
P[A](1)
UP(A)(1),
For any object E, we have an obvious isomorphism S(P,A; E) � P[A](1)⊗E
and we obtain
S(P, S(P,A); S(P,A; E)) � P[S(P,A)](1) ⊗ P[A](1) ⊗ E.
Thus the evaluation morphism of a representation ρ : S(P,A; E) → E
is equivalent to a morphism ρ : P[A](1) ⊗ E → E. One checks further, by
a straightforward inspection, that ρ :S(P,A; E) → E satisfies the unit and
associativity relation of representations if and only if the equivalent morphism
ρ : P[A](1) ⊗ E → E equalizes the morphisms d0,d1 : P[S(P,A)](1) → P[A](1)
and induces a unitary and associative action of UP(A)(1) on E. The conclusion
follows. ⊓⊔
As a corollary, we obtain:
4.3.3 Proposition. The forgetful functor U : RP(A) →E has a left adjoint
F : E→RP(A) which associates to any object X ∈Ethe free left UP(A)module
F (X) =UP(A) ⊗ X. ⊓⊔
In other words, the object F (X) =UP(A) ⊗ X represents the free object
generated by X in the category of representations RP(A).
In the remainder of the section, we determine the operadic enveloping algebras
of commutative, associative and Lie algebras. To simplify, we assume
E = C = k Mod and we use the pointwise representation of tensors in proofs.
As explained in the introduction of this chapter, we can use the principle of
generalized point-tensors (see §0.5) to extend our results to commutative (respectively,
associative, Lie) algebras in categories of dg-modules, in categories
of Σ∗-objects, and in categories of right modules over operads.
4.3.4 Proposition. Let P = F(M)/(R) be an operad in k-modules defined by
generators and relations. The enveloping algebra of any P-algebra A is generated
by formal elements ξ(x1,a1,...,an), whereξ ∈ M(1 + n) ranges over
generating relations of P, together with the relations wρ(x1,a1,...,an) ≡ 0,
where ρ ∈ R(n) and w ∈ Σn+1. ⊓⊔
The next propositions are easy consequences of this statement.
4.3.5 Proposition. For a commutative algebra without unit A, the operadic
enveloping algebra UC(A) is isomorphic to A+, the unitary algebra such that
A+ = 1 ⊕A. ⊓⊔