Free and cofree Hopf algebrasloic.foissy.free.fr/pageperso/expose1.pdf · for any graded set D, one can construct the Hopf algebra of plane rooted trees decorated by D. It is free

IntroductionStructure of a free and cofree Hopf algebra

Free and cofree Hopf algebras

Loïc Foissy

June 2011

Loïc Foissy Free and cofree Hopf algebras


Examples in combinatoricsExamples in physicsExamples from operadsKnown results on these objects

The Hopf algebra FQSym (or Malvenuto-Reutenauer Hopfalgebra) is based on permutations.

The product is given by shifted shuffles :

(12)(123) = (12)(345)

= (12345) + (13245) + (13425) + (13452)

+(31245) + (31425) + (31452) + (34125)

+(34152) + (34512).

Its coproduct is given by standardization :

∆(1432) = (1423)⊗ 1 + (143)⊗ (2)

+(14)⊗ (32) + (1)⊗ (432) + 1⊗ (1432)

= (1423)⊗ 1 + (132)⊗ (1)

+(12)⊗ (21) + (1)⊗ (321) + 1⊗ (1432).




Known resultsFQSym is graded by the length of permutations.It is freely generated by the indecomposablepermutations : σ ∈ Sn is decomposable if there exists1 ≤ i ≤ n − 1 such that σ({1, . . . , i}) = {1, . . . , i}.It is self-dual, via the Hopf pairing defined by〈σ, τ〉 = δσ,τ−1 .So it is also cofree.




Other examplesThe Hopf algebra of parking functions(Novelli-Thibon).The Hopf algebra of uniform block permutations(Aguiar-Orellana).The Hopf algebra of set composition (Aguiar-Mahajan,Bergeron-Zabrocki).




The Hopf algebra of plane rooted trees HPR is anoncommutative model of the Connes-Kreimer Hopf algebraused for Renormalization.

The set of plane rooted forests is a basis of HPR :

1, q , q q , qq , q q q , qq q , q qq , q∨qq, qqq ,

q q q q , qq q q , q qq q , q q qq , qq qq , q∨qq q , q q∨qq, qqq q , q qqq , q∨qq q

, q∨qqq, q∨qq q

,q∨qq q , qqqq . . .

The product is the concatenation of forests.




The coproduct is given by admissible cuts :

∆(t) =∑

c admissible cutPc(t)⊗ Rc(t).

cut c q∨qqq q∨qqq q∨qqq q∨qqq q∨qqq q∨qqq q∨qqq q∨qqqtotal

Admissible ? yes yes yes yes no yes yes no yes

W c(t) q∨qqq qq qq q q∨qq qqq q q q qq qq q q qq q q q q q q q∨qqqRc(t) q∨qqq qq q∨qq qqq × q qq × 1

Pc(t) 1 qq q q × qq q q q × q∨qqq∆( q∨qqq

) = q∨qqq⊗1+1⊗ q∨qqq

+ qq ⊗ qq + q⊗ q∨qq+ q⊗ qqq + qq q⊗ q + q q⊗ qq .




Known resultsThis Hopf algebra is graded by the number of vertices ofthe forests. Its formal series is :

1−√

1− 4X2X

.

It is freely generated by the set of plane rooted trees.It has a non degenerate Hopf pairing, so it is self-dual.So it is also cofree.




Other examplesfor any graded set D, one can construct the Hopf algebraof plane rooted trees decorated by D. It is free and cofreeand its formal series is :

1−√

1− 4D(X )

2D(X )

where D(X ) is the formal series of D.The Hopf algebra of ordered forests.The Hopf algebra of double posets(Malvenuto-Reutenauer).K [X ], if the characteristic of K is zero.




A 2-As Hopf algebra has two associative products with thesame unit, and a coassociative coproduct, with the followingcompatibilities :

1 ∆(xy) = ∆(x)∆(y) : with the first product, it is a Hopfalgebra.

2 ∆(x ? y) = (x ⊗ 1) ? ∆(y) + ∆(x) ? (1⊗ y)− x ⊗ y : withthe second coproduct, it is an infinitesimal Hopf algebra.




The free 2-As algebra on one generator is Hopf. It can bedescribed in words of reduced rooted plane trees, with adecoration on the root.

q∨qq. q = q∨qq q

, q∨qq? q = q∨qq∨qq

∗, q∨qq∗. q = q∨qq∨qq

, q∨qq∗ ? q = q∨qq q

∗

q∨qq. q∨qq

= q∨qq�Hq q

, q∨qq∗. q∨qq

∗ = q�Hq q∨ ∨q qq q, q∨qq

? q∨qq= q�Hq q∨ ∨q qq q

∗, q∨qq∗? q∨qq

∗ = q∨qq�H

q q∗




TheoremThe free 2-As algebra on one generator is freely generated(as an algebra) by q and the trees with their root decoratedby ∗.It is cofree (as it is also an infinitesimal Hopf algebra).




Other examplesThe free dendriform algebra on one generator.The free dipterous algebra on one generator.




ResultsAll these Hopf algebras are self-dual.If two of these Hopf algebras have the same formal series,then they are isomorphic (as graded Hopf algebras).The Lie algebras of primitive elements of these Hopfalgebras are free if the characteristic of the base field iszero.Excepting K [X ], all these objects are isomorphic to a Hopfalgebra of decorated plane trees.

Methods :Construction of explicit isomorphisms or Hopf pairings.(co)associativity splitting.




A bidendriform bialgebra is a Hopf algebra such that theproduct can be written m =≺ + �, and the coproduct can bewritten ∆ = ∆≺ + ∆�, with the following compatibilites :

Dendriform algebra :

(a ≺ b) ≺ c = a ≺ (bc)

(a � b) ≺ c = a � (b ≺ c)

(ab) � c = (a � b) � c.

Dendriform coalgebra :

(∆≺ ⊗ Id) ◦∆≺ = (Id ⊗∆) ◦∆≺

(∆� ⊗ Id) ◦∆≺ = (Id ⊗∆≺) ◦∆�

(∆⊗ Id) ◦∆≺ = (Id ⊗∆�) ◦∆�.




Compatibilites products/coproducts :

∆�(a � b) = a′b′� ⊗ a′′ � b′′� + a′ ⊗ a′′ � b + b′� ⊗ a � b′′�+ab′� ⊗ b′′� + a⊗ b,

∆�(a ≺ b) = a′b′� ⊗ a′′ ≺ b′′� + a′ ⊗ a′′ ≺ b + b′� ⊗ a ≺ b′′�,

∆≺(a � b) = a′b′≺ ⊗ a′′ � b′′≺ + ab′≺ ⊗ b′′≺+b′≺ ⊗ a � b′′≺,

∆≺(a ≺ b) = a′b′≺ ⊗ a′′ ≺ b′′≺ + a′b ⊗ a′′

+b′≺ ⊗ a ≺ b′′≺ + b ⊗ a.




Rigidity theorem

The free dendriform algebras are the (dual of) the Hopfalgebras of decorated plane trees ; they are bidendriformbialgebras.Any graded, connected bidendriform bialgebra is freelygenerated (as a dendriform algebra) by the elements,primitive for both coproducts ∆≺ and ∆� (so is isomorphicto a Hopf algebra of plane decorated trees).




For example, FQSym is bidendriform :

(12) ≺ (123) = (13452) + (31452) + (34152) + (34512)

(12) � (123) = (12345) + (13245) + (13425)

+(31245) + (31425)(34125).

∆≺(1432) = (1423)⊗ 1 + (132)⊗ (1) + (12)⊗ (21)

∆�(1432) = (1)⊗ (321) + 1⊗ (1432).




QuestionsIs a free and cofree Hopf algebra always self-dual ?Are two free and cofree Hopf algebras with the sameformal series always isomorphic ?What can be said on the Lie algebra of primitive elementsof a free and cofree Hopf algebra ?Excepting K [X ], is a free and cofree Hopf algebra alwaysisomorphic to a Hopf algebra of decorated plane trees ?



Self duality and isomorphismsPrimitive elementsPossible formal seriesNon graded isomorphisms

Let H be a graded, connected Hopf algebra. Let H+ be itsaugmentation ideal, g the Lie algebra of its primitive elements.H can be decomposed into four graded subspaces :

H = (g ∩ H+2)⊕m ⊕ h ⊕ w ,

with :1 g = g ∩ H+2 ⊕ h.2 H+2 = g ∩ H+2 ⊕m .

g/g ∩ H+2 ≈ h is the space of indecomposable primitiveelements.

Lemma1 If H is free, h ⊕ w freely generates H.2 If H is free and cofree, g ∩ H+2 and w have the same

formal series.




If H has a symmetric, non degenerate Hopf pairing, then :

g⊥ = (1)⊕ H+2.

Consequently :

Lemma1 g

g∩H+2 ≈ h inherits a non-degenerate pairing.

2 If h and m are chosen, one can choose w such that in anadapted basis to the preceding decomposition, the pairinghas the form :

0 0 0 I0 A 0 00 0 B 0I 0 0 0

.




In the other sense, if H is a free and cofree Hopf algebra, if wefix a symmetric, non-degenerate pairing on h, we can extend it(not uniquely) in a symmetric, non-degenerate Hopf pairing onH.

TheoremAny free and cofree Hopf algebra is self-dual.




Idea of the proof

The pairing is inductively defined on Hn. It is uniquely definedby :

〈x , yz〉 = 〈∆(x), y ⊗ z〉 for all x ∈ w ⊕m, for all y , z ∈ H+.〈x , y〉 = 0 if x , y ∈ w .




Let two free and cofree Hopf algebras H and H ′, with the sameformal series.

1 Then gg∩H+2 and g′

g′∩H′+2 have the same formal series.

2 One can fix isometric non-degenerate pairings on thesetwo spaces.

3 Identifying these spaces with h and h′, one extend thesepairing into Hopf pairing on H and H ′.

4 The isometry between h and h′ is extended into anisometric algebra morphism between H and H ′.

5 Because it is isometric, this algebra morphism is a Hopfalgebra isomorphism.




TheoremTwo free, cofree Hopf algebras are isomorphic as graded Hopfalgebras if, and only if, they have the same formal series.




We now assume that the characteristic of the base field is zero.

Proposition1 g ∩ H+2 = [g, g].2 h freely generates the Lie algebra g and the (free)

subalgebra generated by h is isomorphic to U(g).

Idea of the proof. Uses the Lie algebra and the Lie coalgebra ofH and H∗, and their abelianization.




Consequently, one can determine the formal series of h fromthe formal series of H (and conversely). We put rn = dim(Hn),pn = dim(gn) and sn = dim(hn). Then :

Theorem ∑pnhn = 1− 1∑

rnhn,

∑snhn = 1−

∏(1− hn)pn .




Examples

s1 = r1

s2 = r2 −32

r21 +

12

r1

s3 = r3 +13

r1 − 3r1r2 −12

r21 +

136

r31




TheoremLet R(X ) =

∑rnX n a formal series in N[[X ]]. There exists a

free and cofree Hopf algebra of formal series R(X ) if, and onlyif, sn ≥ 0 for all n.

Idea of the proof. =⇒. As sn is the number of generators of g ofdegree n.

⇐=. Such a Hopf algebra is constructed a quotient/subalgebraof a Hopf algebra of decorated plane trees with sn generators indegree n for all n.




s1 s2 s3 s4 s5 s6 s7HPRT 1 1 1 3 7 24 72

2-As(1) 1 1 2 8 31 141 642FQSym 1 1 2 10 55 377 2 892

NCQSym 1 2 6 39 305 2 900 31 460PQSym 1 2 9 80 901 12 564 206 476

HUBP 1 2 9 86 1 083 17 621 353 420HDP 1 2 12 165 3 545 116 621 5 722 481




The coefficients dn are defined by :

∑dnX n =

∑rnX n − 1(∑

rnX n)2 .

Examples

d1 = r1

d2 = r2 − 2r21

d3 = r3 − 4r2r1 + 3r21 .




PropositionThere exists a Hopf algebra of decorated plane trees of formalseries

∑rnX n if, and only if, dn ≥ 0 for all n.

CorollaryThere exists free and cofree Hopf algebras, not isomorphic toK [X ], nor to any Hopf algebra of decorated plane trees. (Forexample, take s1 = 1, s2 = 1, s3 = 0).




d1 d2 d3 d4 d5 d6 d7HPRT 1 0 0 0 0 0 0

2-As(1) 1 0 1 4 17 76 353FQSym 1 0 1 6 39 284 2 305

NCQSym 1 1 4 28 240 2 384 26 832PQSym 1 1 7 66 786 11 278 189 391

HUBP 1 1 7 72 962 16 135 330 624HDP 1 1 10 148 3 336 112 376 5 591 196




TheoremTwo free and cofree Hopf algebras H and H ′ are isomorphic asnon graded Hopf algebras if, and only if,dim

(g

[g,g]

)= dim

(g′

[g′,g′]

).

Corollary

Except K [X ], all the Hopf algebras of the introduction areisomorphic (as non graded Hopf algebras).




Idea of the proof.

1 The gradation of H and H ′ is refined into a bigradation of Hand H ′ :

Hn =∞⊕

k=0

Hn,k .

2 So H and H ′ inherit a second gradation :

Hk =∞⊕

n=0

Hn,k .

3 For a good choice of the bigradation, h and h′ have thesame formal series for the second gradation.

4 So H and H ′ are isomorphic as graded Hopf algebras, forthe second gradation.


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Free and cofree Hopf algebrasloic.foissy.free.fr/pageperso/expose1.pdf · for any graded set D, one can construct the Hopf algebra of plane rooted trees decorated by D. It is free