Symmetric Monoidal Categories for Operads - Index of

4.3 Enveloping Algebras 87
4.2.4 Examples: The Case of Classical Operads in k-Modules. In the
usual examples of operads in k-modules, P = C, L, A, the equivalence of §4.2.3
returns the following assertions:
(a) Let A be any commutative algebra in k-modules. The structure of a
representation of A is determined by morphisms μ : E ⊗ A → E and μ :
A ⊗ E → E so that we have the symmetry relation μ(a, x) =μ(x, a), for all
a ∈ A, x ∈ E, and the associativity relations
μ(μ(a, b),x)=μ(a, μ(b, x)), μ(μ(a, x),b)=μ(a, μ(x, b)),
μ(μ(x, a),b)=μ(x, a · b),
for all a, b ∈ A, x ∈ E, where we also use the notation μ(a, b) =a·b to denote
the product of A. As a consequence, the category of representations RC(A),
where A is any commutative algebra, is isomorphic to the usual category of
left A-modules.
(b) Let A be any associative algebra in k-modules. The structure of a representation
of A is determined by morphisms μ : E⊗A → E and μ : A⊗E → E
so that we have the associativity relations
μ(μ(a, b),x)=μ(a, μ(b, x)), μ(μ(a, x),b)=μ(a, μ(x, b)),
μ(μ(x, a),b)=μ(x, μ(a, b)),
for all a, b ∈ A, x ∈ E, where we also use the notation μ(a, b) =a·b to denote
the product of A. Hence the category of representations RA(A), where A is
any associative algebra, is isomorphic to the usual category of A-bimodules.
(c) Let G be any Lie algebra in k-modules. The structure of a representation
of G is determined by morphisms γ : E ⊗ G → E and γ : G ⊗E → E so that
we have the antisymmetry relation γ(g, x) =γ(x, g), for all g ∈ G, x ∈ E,
and the Jacobi relations
γ(γ(g, h),x)+γ(γ(h, x),g)+γ(γ(x, g),h)=0
for all g, h ∈ G, x ∈ E, where we also use the notation γ(g, h) =[g, h]todenote
the Lie bracket of G. Hence the category of representations RL(G), where G is
any Lie algebra, is isomorphic to the usual category of representations of the
Lie algebra G.
4.3 Enveloping Algebras
The enveloping algebra of a P-algebra A is the term UP(A)(1) of the enveloping
operad UP(A). The first purpose of this section is to prove that
the category of representations of A is isomorphic to the category of left
UP(A)(1)-modules. Then we determine the enveloping algebra of algebras