Introduction Structure of a free and cofree Hopf algebra
Free and cofree Hopf algebras Loïc Foissy
June 2011
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
The Hopf algebra FQSym (or Malvenuto-Reutenauer Hopf algebra) 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). Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Known results FQSym is graded by the length of permutations. It is freely generated by the indecomposable permutations : σ ∈ Sn is decomposable if there exists 1 ≤ i ≤ n − 1 such that σ({1, . . . , i}) = {1, . . . , i}. It is self-dual, via the Hopf pairing defined by hσ, τ i = δσ,τ −1 . So it is also cofree.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Other examples The 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).
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
The Hopf algebra of plane rooted trees HPR is a noncommutative model of the Connes-Kreimer Hopf algebra used for Renormalization. The set of plane rooted forests is a basis of HPR : q q q q qq q 1, q , q q , q , q q q , q q , q q , ∨q , q , q q qq qq q qq q q qq ∨ qq q q qq q q q q q q q q q , q q q , q q q , q q q , q q , ∨q q , q ∨q , q q , q q , ∨q , ∨q , ∨q , qq ,
The product is the concatenation of forests.
Loïc Foissy
Free and cofree Hopf algebras
q qq q ...
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
The coproduct is given by admissible cuts : X ∆(t) = P c (t) ⊗ R c (t). c admissible cut q q q ∨q
q qq ∨q
cut c Admissible ? yes yes
R c (t)
q q q ∨q q q q ∨q
P c (t)
1
W c (t)
q qq ∨q
q qq ∨q
q qq ∨q
yes
yes
qq qq
qq q ∨q
qq
∨q
qq qq qq q
qq
q
q
qq
q qq ∨q
q q q ∨q
q q q ∨q
total yes
no
yes yes
no
q q qq
qq q q
qq q q
qqqq
q q q ∨q
×
q
qq
×
1
×
qq q
qq
×
∨q
q q q
q q q qq q qq q q qq qq q q q ∆( ∨q ) = ∨q ⊗1+1⊗ ∨q + q ⊗ q + q ⊗ ∨q + q ⊗ q + q q ⊗ q + q q ⊗ q . Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Known results This Hopf algebra is graded by the number of vertices of the forests. Its formal series is : √ 1 − 1 − 4X . 2X 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.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Other examples for any graded set D, one can construct the Hopf algebra of plane rooted trees decorated by D. It is free and cofree and its formal series is : p 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.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
A 2-As Hopf algebra has two associative products with the same unit, and a coassociative coproduct, with the following compatibilities : 1
∆(xy ) = ∆(x)∆(y ) : with the first product, it is a Hopf algebra.
2
∆(x ? y ) = (x ⊗ 1) ? ∆(y ) + ∆(x) ? (1 ⊗ y ) − x ⊗ y : with the second coproduct, it is an infinitesimal Hopf algebra.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
The free 2-As algebra on one generator is Hopf. It can be described in words of reduced rooted plane trees, with a decoration on the root. qq qq qqq q q ∨q q ∨q . q = ∨q , ∨q ? q = ∨q ∗ , q q q q qq qq q q q q q q qq q q∨q ∨q . ∨q = H ∨ q , ∨q ∗ . ∨q ∗ =∨H ,
Loïc Foissy
qq qq ∨q q q q qqq ∨q ∗ . q = ∨q , ∨q ∗ ? q = ∨q ∗ qq qq q q q q q q q q q q q ∨q ? ∨q =∨qHq∨q∗ , ∨q ∗ ? ∨q ∗ = qH∨ q ∗
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Theorem The free 2-As algebra on one generator is freely generated (as an algebra) by q and the trees with their root decorated by ∗. It is cofree (as it is also an infinitesimal Hopf algebra).
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Other examples The free dendriform algebra on one generator. The free dipterous algebra on one generator.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Results All 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 Hopf algebras are free if the characteristic of the base field is zero. Excepting K [X ], all these objects are isomorphic to a Hopf algebra of decorated plane trees. Methods : Construction of explicit isomorphisms or Hopf pairings. (co)associativity splitting. Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
A bidendriform bialgebra is a Hopf algebra such that the product can be written m =≺ + , and the coproduct can be written ∆ = ∆≺ + ∆ , 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 ⊗ ∆ ) ◦ ∆ . Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Compatibilites products/coproducts : 0 00 0 00 ∆ (a b) = a0 b ⊗ a00 b + a0 ⊗ a00 b + b ⊗ a b 0 00 +ab ⊗ b + a ⊗ b, 0 00 0 00 ∆ (a ≺ b) = a0 b ⊗ a00 ≺ b + a0 ⊗ a00 ≺ b + b ⊗ a ≺ b , 0 00 0 00 ∆≺ (a b) = a0 b≺ ⊗ a00 b≺ + ab≺ ⊗ b≺ 0 00 +b≺ ⊗ a b≺ , 00 0 + a0 b ⊗ a00 ∆≺ (a ≺ b) = a0 b≺ ⊗ a00 ≺ b≺ 0 00 ⊗ a ≺ b≺ + b ⊗ a. +b≺
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Rigidity theorem The free dendriform algebras are the (dual of) the Hopf algebras of decorated plane trees ; they are bidendriform bialgebras. Any graded, connected bidendriform bialgebra is freely generated (as a dendriform algebra) by the elements, primitive for both coproducts ∆≺ and ∆ (so is isomorphic to a Hopf algebra of plane decorated trees).
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
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).
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Examples in combinatorics Examples in physics Examples from operads Known results on these objects
Questions Is a free and cofree Hopf algebra always self-dual ? Are two free and cofree Hopf algebras with the same formal series always isomorphic ? What can be said on the Lie algebra of primitive elements of a free and cofree Hopf algebra ? Excepting K [X ], is a free and cofree Hopf algebra always isomorphic to a Hopf algebra of decorated plane trees ?
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Let H be a graded, connected Hopf algebra. Let H + be its augmentation 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 primitive elements. Lemma 1 2
If H is free, h ⊕ w freely generates H. If H is free and cofree, g ∩ H +2 and w have the same formal series. Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
If H has a symmetric, non degenerate Hopf pairing, then : g⊥ = (1) ⊕ H +2 . Consequently : Lemma 1 2
g g∩H +2
≈ h inherits a non-degenerate pairing.
If h and m are chosen, one can choose w such that in an adapted basis to the preceding decomposition, the pairing has the form : 0 0 0 I 0 A 0 0 0 0 B 0 . I 0 0 0 Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
In the other sense, if H is a free and cofree Hopf algebra, if we fix a symmetric, non-degenerate pairing on h, we can extend it (not uniquely) in a symmetric, non-degenerate Hopf pairing on H. Theorem Any free and cofree Hopf algebra is self-dual.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Idea of the proof
The pairing is inductively defined on Hn . It is uniquely defined by : hx, yzi = h∆(x), y ⊗ zi for all x ∈ w ⊕ m, for all y , z ∈ H + . hx, y i = 0 if x, y ∈ w.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Let two free and cofree Hopf algebras H and H 0 , with the same formal series. g g∩H +2
and
g0 g0 ∩H 0+2
have the same formal series.
1
Then
2
One can fix isometric non-degenerate pairings on these two spaces.
3
4
5
Identifying these spaces with h and h0 , one extend these pairing into Hopf pairing on H and H 0 . The isometry between h and h0 is extended into an isometric algebra morphism between H and H 0 . Because it is isometric, this algebra morphism is a Hopf algebra isomorphism.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Theorem Two free, cofree Hopf algebras are isomorphic as graded Hopf algebras if, and only if, they have the same formal series.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
We now assume that the characteristic of the base field is zero. Proposition 1 2
g ∩ H +2 = [g, g]. 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 of H and H ∗ , and their abelianization.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Consequently, one can determine the formal series of h from the formal series of H (and conversely). We put rn = dim(Hn ), pn = dim(gn ) and sn = dim(hn ). Then : Theorem X
X
pn hn = 1 − X
sn hn = 1 −
Loïc Foissy
1 rn h n
,
Y (1 − hn )pn .
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Examples s 1 = r1 1 3 s2 = r2 − r12 + r1 2 2 13 3 1 1 s3 = r3 + r1 − 3r1 r2 − r12 + r 3 2 6 1
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Theorem P Let R(X ) = rn X n a formal series in N[[X ]]. There exists a free and cofree Hopf algebra of formal series R(X ) if, and only if, sn ≥ 0 for all n. Idea of the proof. =⇒. As sn is the number of generators of g of degree n. ⇐=. Such a Hopf algebra is constructed a quotient/subalgebra of a Hopf algebra of decorated plane trees with sn generators in degree n for all n.
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
s1 s2 s3 s4 s5 s6 s7 HPRT 1 1 1 3 7 24 72 2-As(1) 1 1 2 8 31 141 642 FQSym 1 1 2 10 55 377 2 892 NCQSym 1 2 6 39 305 2 900 31 460 PQSym 1 2 9 80 901 12 564 206 476 HUBP 1 2 9 86 1 083 17 621 353 420 HDP 1 2 12 165 3 545 116 621 5 722 481
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
The coefficients dn are defined by : X X rn X n − 1 n dn X = X 2 . n rn X Examples d 1 = r1 d2 = r2 − 2r12 d3 = r3 − 4r2 r1 + 3r12 .
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Proposition There exists a Hopf algebra of decorated plane trees of formal P series rn X n if, and only if, dn ≥ 0 for all n. Corollary There exists free and cofree Hopf algebras, not isomorphic to K [X ], nor to any Hopf algebra of decorated plane trees. (For example, take s1 = 1, s2 = 1, s3 = 0).
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
d1 d2 d3 d4 d5 d6 d7 HPRT 1 0 0 0 0 0 0 2-As(1) 1 0 1 4 17 76 353 FQSym 1 0 1 6 39 284 2 305 NCQSym 1 1 4 28 240 2 384 26 832 PQSym 1 1 7 66 786 11 278 189 391 HUBP 1 1 7 72 962 16 135 330 624 HDP 1 1 10 148 3 336 112 376 5 591 196
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Theorem Two free and cofree Hopf algebras H and H 0 are isomorphic as non graded Hopf algebras 0 if, and only if, g dim [g,g] = dim [g0g,g0 ] . Corollary Except K [X ], all the Hopf algebras of the introduction are isomorphic (as non graded Hopf algebras).
Loïc Foissy
Free and cofree Hopf algebras
Introduction Structure of a free and cofree Hopf algebra
Self duality and isomorphisms Primitive elements Possible formal series Non graded isomorphisms
Idea of the proof. 1
The gradation of H and H 0 is refined into a bigradation of H and H 0 : ∞ M Hn,k . Hn = k =0
2
So H and
H0
inherit a second gradation : k
H =
∞ M
Hn,k .
n=0 3
4
For a good choice of the bigradation, h and h0 have the same formal series for the second gradation. So H and H 0 are isomorphic as graded Hopf algebras, for the second gradation. Loïc Foissy
Free and cofree Hopf algebras
