We have hitherto not really used the antipode of a Hopf algebra; thus, we could just as well have replaced the words “Hopf algebra” by “bialgebra” through-out the entire preceding text23. Let us now connect the preceding results with antipodes.
The antipode of any Hopf algebra H will be denoted by SH.
Proposition 6.1. Let k be a commutative ring. Let A be a commutative k-algebra. Let H be ak-Hopf algebra. Let G be an A-Hopf algebra. Then, every k-algebra homomorphism f : H → G which is a (k,A)-coalgebra homomor-phism must also satisfy f ◦SH =SG◦ f.
Proof of Proposition 6.1. We know that H is a k-Hopf algebra. Thus, A⊗H is an A-Hopf algebra. Its definition by extending scalars yields that its antipode is given bySA⊗H =idA⊗SH.
Let f : H → G be a k-algebra homomorphism which is a (k,A)-coalgebra homomorphism. Then, f] : A⊗H → Gis an A-coalgebra homomorphism (since f is a (k,A)-coalgebra homomorphism) and an A-algebra homomorphism (by Proposition 3.4). Hence, f] is an A-bialgebra homomorphism, thus an A-Hopf algebra homomorphism (since every A-bialgebra homomorphism between two A-Hopf algebras is an A-Hopf algebra homomorphism). Thus, f] commutes with the antipodes, i.e., satisfies f]◦SA⊗H =SG◦ f].
Now, letιbe the canonicalk-module homomorphismH → A⊗H, h7→1⊗h.
Then,(idA⊗SH)◦ι=ι◦SH. On the other hand, f]◦ι = f (this is easy to check).
Thus, f
|{z}
=f]◦ι
◦SH = f]◦ ι◦SH
| {z }
=(idA⊗SH)◦ι
= f]◦(idA⊗SH)
| {z }
=SA⊗H
◦ι= f]◦SA⊗H
| {z }
=SG◦f]
◦ι
=SG◦ f] ◦ι
| {z }
=f
=SG◦ f.
This proves Proposition 6.1.
Corollary 6.2. Let k be a commutative ring. Let H be a commutative con-nected graded k-Hopf algebra. Define a map βH : H → H⊗QSymk as in Definition 5.1. Then,
βH◦SH =idH⊗SQSymk
◦βH.
23That said, we would not have gained anything this way, because any connected graded k-bialgebra is ak-Hopf algebra (see [GriRei14, Proposition 1.36]).
Proof of Corollary 6.2. Theorem 5.7 (a) shows that the map βH is a k-algebra ho-momorphism H →H⊗QSymkand a graded (k,H)-coalgebra homomorphism.
Thus, Proposition 6.1 (applied to A = H, G = H⊗QSymk and f = βH) shows that βH◦SH =SH⊗QSymk ◦βH.
But the H-Hopf algebra H⊗QSymk is defined by extension of scalars; thus, its antipode is given bySH⊗QSymk =idH⊗SQSym
k. Hence, βH◦SH = SH⊗QSymk
| {z }
=idH⊗SQSym
k
◦βH =idH⊗SQSym
k
◦βH.
This proves Corollary 6.2.
Corollary 6.3. Let k be a commutative ring. Let H be a commutative con-nected graded k-Hopf algebra. Define a map βH : H → H⊗QSymk as in Definition 5.1. Then,
SH =idH⊗εP◦SQSym
k
◦βH.
Proof of Corollary 6.3. We have
idH⊗εP◦SQSymk
| {z }
=(idH⊗εP)◦(idH⊗SQSym
k)
◦βH
= (idH⊗εP)◦idH⊗SQSymk
◦βH
| {z }
=βH◦SH (by Corollary 6.2)
= (idH⊗εP)◦βH
| {z }
=id (by Theorem 5.7(b))
◦SH =SH,
and thus Corollary 6.3 is proven.
Remark 6.4. What I find remarkable about Corollary 6.3 is that it provides a formula for the antipode SH of H in terms of βH and QSymk. Thus, in order to understand the antipode of H, it suffices to study the map βH and the antipode of QSymk well enough.
Similar claims can be made about other endomorphisms of H, such as the Dynkin idempotent or the Eulerian idempotent (whenk is aQ-algebra). Bet-ter yet, we can regard the map βH : H → H⊗QSymk as an “embedding” of thek-Hopf algebraHinto theH-Hopf algebraH⊗QSymk ∼=QSymH. Here, I am using the word “embedding” in scare quotes, since this map is not a Hopf algebra homomorphism (its domain and its target are Hopf algebras over dif-ferent base rings); nevertheless, the map βH is injective (by Theorem 5.7 (b)),
and the corresponding map (βH)] : H⊗H → H⊗QSymk (sending every a⊗h to aβH(h)) is a graded H-Hopf algebra homomorphism (because it is graded, an H-algebra homomorphism and an H-coalgebra homomorphism);
this shows thatβH commutes with various maps defined canonically in terms of a commutative connected graded Hopf algebra. It appears possible to use this for proving identities in commutative connected graded Hopf algebra.
7. Questions
Let me finish with some open-ended questions, which probably are not particu-larly insightful, but (in my opinion) rather natural.
Question 7.1. It is well-known (see, e.g., [GriRei14, §5.3]) that the graded Hopf-algebraic dual of the graded Hopf algebra QSym is a graded Hopf al-gebra NSym. The second comultiplication ∆P and the second counit εP on QSym dualize to a second multiplication mP and a second unit uP on NSym, albeit uP does not really live inside NSym (in fact, it lives in the completion of NSym with respect to its grading). We denote the “almost-k-bialgebra”
(NSym,mP,uP,∆,ε)(“almost” because uP ∈/NSym) by NSym2.
We can more or less dualize Theorem 5.7. As a result, instead of a QSym2 -comodule structure on every commutative graded connected Hopf algebra H, we obtain an NSym2-module structure on every cocommutative graded connected Hopf algebra H. This structure is rather well-known: (I believe) it has Hα ∈ NSym2 act as the convolution product πa1?πa2 ?· · ·?πak ∈ EndH for every composition α = (a1,a2, . . . ,ak) (where ? denotes the convolution product in EndH). This should be somewhere in the papers by Patras and Reutenauer on descent algebras; it is essentially the way to transfer informa-tion from thedescent algebra NSym2 toadescent algebra EndgradedH,◦ of a cocommutative graded connected Hopf algebra H.
Is it possible to prove that this works using universal properties like I have done above for Theorem 5.7? (Just saying “dualize Theorem 5.7” is not enough, because dualization over arbitrary commutative rings is a heuristic, not a proof strategy; there does not seem to be a general theorem stating that
“the dual of a correct result is correct”, at least when the result has assump-tions about gradedness and similar things.)
If the answer is positive, can we use this to give a slick proof of Solomon’s Mackey formula? (I am not saying that there is need for slick proofs of this formula – not after those by Gessel and Bidigare –, but it would be interest-ing to have a new one. I am thinkinterest-ing of lettinterest-ing both NSym2 and the sym-metric groups act on the tensor algebra T(V) of an infinite-dimensional free k-module V; one then only needs to check that the actions match.)
Note that if u and v are two elements of NSym2, then the action of the NSym-product uv (not the NSym2-product!) on H is the convolution of the
actions of u and v. So the action map NSym2 →EndH takes the multiplica-tion of NSym2to composition, and the multiplication of NSym to convolution.
Question 7.2. In Question 7.1, we found a k-algebra homomorphism