nLab
primitive element

Idea

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; then an element xCx\in C 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.

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:

Proposition

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:

Proposition

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.)

Revised on July 30, 2014 15:37:54 by Todd Trimble (67.81.95.215)