nLab primitive element


(This article is about primitive elements in coalgebra theory, not about primitive elements for finite field extensions.)



Working over a commutative ring RR, recall that a unital or coaugmented coalgebra is an RR-coalgebra (C,Δ:CC RC,ϵ:CR)(C, \Delta: C \to C \otimes_R C, \epsilon: C \to R) equipped with a coalgebra map u:RCu: R \to C. Abusing notation, denote u(1)Cu(1) \in C by 11.

Primitive element in a coalgebra

An element xx in a coalgebra CC is primitive if Δ(x)=1x+x1\Delta(x) = 1\otimes x + x\otimes 1. This condition implies ϵ(x)=0\epsilon(x) = 0.

This notion generalizes straightforwardly to unital corings over RR and even more generally to any unital comonoid object in a CMon-enriched monoidal category (see below).

Primitive element in a comodule

Let NN be a comodule over CC, with co-action map Ψ:NC RN\Psi \colon N \longrightarrow C \otimes_R N. Then an element nNn \in N is called a primitive element or coinvariant if Ψ(n)=1n\Psi(n) = 1 \otimes n.

The RR-module prim(N)prim(N) of primitive elements of NN is naturally to

  1. the module of comodule homomorphisms out of RR into NN

    prim(N)Hom C(R,N). prim(N) \simeq Hom_{C}(R,N) \,.
  2. the cotensor product of NN with AA

    prim(N)A RN. prim(N) \simeq A \Box_R N \,.

Object of primitive elements of a unital comonoid in a CMonCMon-enriched monoidal category

Let (C,Δ,ϵ)(C,\Delta,\epsilon) be a unital comonoid object in a CMon-enriched monoidal category (𝒞,,I)(\mathcal{C},\otimes,I). The object PP of primitive elements of CC is defined if it exists, as the equalizer in this diagram:


If the unital comonoid CC admits an object PP of primitive elements, the commutative rig 𝒞[I,I]\mathcal{C}[I,I] verifies 212 \neq 1, and the homsets 𝒞\mathcal{C}[A,B] are multiplicatively cancellative modules over 𝒞[I,I]\mathcal{C}[I,I], we then have:


In this diagram, the path from CC to CC by the upper square (with below side equal to Δ\Delta) is equal to the identity and the two paths from PP to CCC \otimes C are equal:

We thus have that:

i=i;Δ;1ϵ=i;(1η+η1);(1ϵ)i = i;\Delta;1 \otimes \epsilon = i;(1 \otimes \eta + \eta \otimes 1);(1 \otimes \epsilon)


i;ϵ=i;Δ;1ϵ;ϵ=i;(1η+η1);(1ϵ);ϵ i;\epsilon = i;\Delta;1 \otimes \epsilon;\epsilon = i;(1 \otimes \eta + \eta \otimes 1);(1 \otimes \epsilon);\epsilon
=i;(1η);(1ϵ);ϵ+i;(η1);(1ϵ);ϵ = i;(1 \otimes \eta);(1 \otimes \epsilon);\epsilon + i;(\eta \otimes 1);(1 \otimes \epsilon);\epsilon

We then have:

𝒞[P,I]\mathcal{C}[P,I] being a multiplicatively cancellative 𝒞[I,I]\mathcal{C}[I,I]-module, if i;ϵ0i;\epsilon \neq 0, we would have that 1=21 = 2 in 𝒞[I,I]\mathcal{C}[I,I] which is false, therefore i;ϵ=0i;\epsilon = 0.


Primitives in a Hopf algebra

A straightforward calculation shows that the module of primitive elements in a Hopf algebra HH (or even in a bialgebra HH) is a Lie subalgebra of the underlying Lie algebra of HH (whose bracket is the algebra commutator). Thus, taking primitive elements yields a functor

P:HopfAlgLieAlgP: HopfAlg \to LieAlg

(and of course we have more generally a functor P:BiAlgLieAlgP: BiAlg \to LieAlg which is an extension along the full inclusion HopfAlgBiAlgHopfAlg \to BiAlg).

For a Lie algebra LL, let U(L)U(L) be its universal enveloping algebra:

U(L)=T(L)/IU(L) = T(L)/I

where II is the two-sided ideal generated by elements of the form xyyx[x,y]x y - y x - [x, y] where x,yLx, y \in L. This carries a bialgebra structure whose comultiplication δ:U(L)U(L)U(L)\delta: U(L) \to U(L) \otimes U(L) is uniquely determined by the rule δ(x)=1x+x1\delta(x) = 1 \otimes x + x \otimes 1 for xLx \in L. Since this says xLx \in L is primitive, the counit ϵ:U(L)R\epsilon: U(L) \to R is forced to be the algebra map such ϵ(x)=0\epsilon(x) = 0 for all xLx \in L, and also the Hopf antipode is uniquely determined: σ(x)=x\sigma(x) = -x for xLx \in L.

The following proposition is entirely straightfoward:


The functor U:LieAlgBiAlgU: LieAlg \to BiAlg is left adjoint to the functor P:BiAlgLieAlgP: BiAlg \to LieAlg.

This result is essentially tautologous and holds for any commutative ring of arbitrary characteristic. (This despite the fact that the U(L)U(L) as defined above is not as well-behaved in nonzero characteristic as one might like; e.g. the PBW theorem fails.) More information on this adjunction may require more restrictive hypotheses:


For R=kR = k a field of characteristic zero, the unit LPU(L)L \to P U(L) is an isomorphism.

An immediate consequence is that for such ground fields kk, the functor U:LieAlgBiAlgU: LieAlg \to BiAlg is fully faithful. Of course, UU lands in the full subcategory of cocommutative Hopf algebras, which is exactly the category of group objects in the cartesian monoidal category of cocommutative coalgebras.

The Milnor-Moore theorem gives further information: for Hopf algebras over a field of characteristic zero, the counit UP(H)HU P(H) \to H is a monomorphism, and an isomorphism in case HH satisfies a suitable conilpotency condition.

(More needs to be added.)

Last revised on November 25, 2023 at 12:18:23. See the history of this page for a list of all contributions to it.