Hopf algebras and Diophantine problems

Academia Sinica, Taipei October 30, 2003

Hopf algebras and

Diophantine problems

Michel Waldschmidt

miw@math.jussieu.fr http://www.math.jussieu.fr/∼miw/

Hopf algebras (commutative, cocommutative, of finite type) Algebraic groups (commutative, linear, over Q)

Exponential polynomials

Transcendence of values of exponential polynomials

Algebra of multizeta values

Algebras (over k = C or k = Q)

A k -algebra (A, m, η ) is a k-vector space A with a product m : A ⊗ A → A and a unit η : k −→ A which are k -linear maps such that the following diagrams commute:

(Associativity)

A ⊗ A ⊗ A

−−→ A ⊗ A

Id⊗m

↓ ↓

A ⊗ A − →

m

A

(Unit)

k ⊗ A

−−→ A ⊗ A

←−− A ⊗ k

↓ ↓

↓

A = A = A

Commutative algebras A k -algebra is commutative if the diagram

A ⊗ A − →

A ⊗ A

m

↓ ↓

A = A

commutes. Here τ (x ⊗ y) = y ⊗ x.

Coalgebras

A k -coalgebra (A, ∆, ) is a k -vector space A with a coproduct

∆ : A → A ⊗ A and a counit : A −→ k which are k-linear maps such that the following diagrams commute:

(Coassociativity)

A − →

A ⊗ A

∆

↓ ↓

A ⊗ A −−→

Id⊗∆

A ⊗ A ⊗ A

(Counit)

A = A = A

↓ ↓

↓

k ⊗ A ←−−

⊗Id

A ⊗ A −−→

Id⊗

A ⊗ k

Commutative coalgebras A k -coalgebra is commutative if the diagram

A = A

∆

↓ ↓

A ⊗ A ← −

τ

A ⊗ A

commutes.

Bialgebras

A bialgebra (A, m, η, ∆, ) is a k -algebra (A, m, η ) together with a coalgebra structure (A, ∆, ) which is compatible: ∆ and are algebra morphisms

∆(xy ) = ∆(x)∆(y), (xy ) = (x)(y).

Hopf Algebras

A Hopf algebra (H, m, η, ∆, , S ) is a bialgebra (H, m, η, ∆, ) with an antipode S : H → H which is a k-linear map such that the following diagram commutes:

H ⊗ H ←

− H − →

H ⊗ H

Id⊗S

↓

↓ ↓

H ⊗ H − →

m

H ← −

m

H ⊗ H

In a Hopf Algebra the primitive elements

∆(x) = x ⊗ 1 + 1 ⊗ x

satisfy (x) = 0 and S (x) = −x; they form a Lie algebra for the bracket

[x, y] = xy − yx.

In a Hopf Algebra the primitive elements

∆(x) = x ⊗ 1 + 1 ⊗ x

satisfy (x) = 0 and S (x) = −x; they form a Lie algebra for the bracket

[x, y] = xy − yx.

The group-like elements

∆(x) = x ⊗ x, x 6= 0

are invertible, they satisfy (x) = 1, S (x) = x

and form a

Example 1.

Let G be a finite multiplicative group, kG the algebra of G over k which is a k vector-space with basis G. The mapping

m : kG ⊗ kG → kG extends the product

(x, y) 7→ xy of G by linearity. The unit

η : k → kG

maps 1 to 1

.

Define a coproduct and a counit

∆ : kG → kG ⊗ kG and : kG → k

by extending

∆(x) = x ⊗ x and (x) = 1 for x ∈ G

by linearity. The antipode

S : kG → kG is defined by

S (x) = x

for x ∈ G.

Since ∆(x) = x ⊗ x for x ∈ G this Hopf algebra kG is cocommutative.

It is a commutative algebra if and only if G is commutative.

The set of group like elements is G: one recovers G from kG.

Example 2.

Again let G be a finite multiplicative group. Consider the k- algebra k

of mappings G → k, with basis δ

(g ∈ G), where

δ

(g

) =

1 for g

= g, 0 for g

6= g.

Define m by

m(δ

⊗ δ

) = δ

δ

.

Hence m is commutative and m(δ

⊗ δ

) = δ

for g ∈ G.

The unit η : k → k

maps 1 to P

g∈G

δ

.

Define a coproduct ∆ : k

→ k

⊗ k

and a counit : k

→ k by

∆(δ

) = X

g⁰g⁰⁰=g

δ

⊗ δ

and (δ

) = δ

(1

).

The coproduct ∆ is cocommutative if and only if the group G is commutative.

Define an antipode S by

S (δ

) = δ

.

Remark. One may identify k

⊗ k

and k

with

δ

⊗ δ

= δ

.

Duality of Hopf Algebras

The Hopf algebras kG from example 1 and k

from example 2 are dual from each other:

kG × k

−→ k (g

, δ

) 7−→ δ

(g

)

The basis G of kG is dual to the basis (δ

)

of k

.

Example 3.

Let G be a topological compact group over C. Denote by R (G) the set of continuous functions f : G → C such that the translates f

: x 7→ f (tx), for t ∈ G, span a finite dimensional vector space.

Define a coproduct ∆, a counit and an antipode S on R (G) by

∆f (x, y) = f (xy ), (f ) = f (1), Sf (x) = f (x

) for x, y ∈ G.

Hence R (G) is a commutative Hopf algebra.

Example 4.

Let g be a Lie algebra, U ( g ) its universal envelopping algebra, namely T ( g )/ I where T ( g ) is the tensor algebra of g and I the two sided ideal generated by XY − Y X − [X, Y ].

Define a coproduct ∆, a counit and an antipode S on U ( g ) by

∆(x) = x ⊗ 1 + 1 ⊗ x, (x) = 0, S (x) = −x for x ∈ g .

Hence U ( g ) is a cocommutative Hopf algebra.

The set of primitive elements is g : one recovers g from U ( g ).

Duality of Hopf Algebras (again)

Let G be a compact connected Lie group with Lie algebra g .

Then the two Hopf algebras R (G) and U ( g ) are dual from each

other.

Bicommutative Hopf algebras of finite type 1.

H = k[X ], ∆(X ) = X ⊗ 1 + 1 ⊗ X , (X ) = 0, S (X ) = −X .

Bicommutative Hopf algebras of finite type 1.

H = k[X ], ∆(X ) = X ⊗ 1 + 1 ⊗ X , (X ) = 0, S (X ) = −X .

k[X ] ⊗ k [X ] ' k[T

, T

], X ⊗ 1 7→ T

, 1 ⊗ X 7→ T

∆P (X ) = P (T

+ T

), P (X ) = P (0), SP (X ) = P (−X ).

Bicommutative Hopf algebras of finite type 1.

H = k[X ], ∆(X ) = X ⊗ 1 + 1 ⊗ X , (X ) = 0, S (X ) = −X .

k[X ] ⊗ k [X ] ' k[T

, T

], X ⊗ 1 7→ T

, 1 ⊗ X 7→ T

∆P (X ) = P (T

+ T

), P (X ) = P (0), SP (X ) = P (−X ).

G

(K ) = Hom

(k[X ], K ), k[G

] = k[X ]

k [G ] is a bicommutative Hopf algebra of finite type.

Bicommutative Hopf algebras of finite type 2.

H = k[Y, Y

], ∆(Y ) = Y ⊗ Y , (Y ) = 1, S(Y ) = Y

.

Bicommutative Hopf algebras of finite type 2.

H = k[Y, Y

], ∆(Y ) = Y ⊗ Y , (Y ) = 1, S(Y ) = Y

.

H ⊗ H ' k [T

, T

, T

, T

], Y ⊗ 1 7→ T

, 1 ⊗ Y 7→ T

∆P (Y ) = P (T

T

), P (Y ) = P (1), SP (Y ) = P (Y

).

Bicommutative Hopf algebras of finite type 2.

H = k[Y, Y

], ∆(Y ) = Y ⊗ Y , (Y ) = 1, S(Y ) = Y

.

H ⊗ H ' k [T

, T

, T

, T

], Y ⊗ 1 7→ T

, 1 ⊗ Y 7→ T

∆P (Y ) = P (T

T

), P (Y ) = P (1), SP (Y ) = P (Y

).

G

(K ) = Hom

(k[Y, Y

], K ), k [G

] = k [Y, Y

],

k [G

] is a bicommutative Hopf algebra of finite type.

Bicommutative Hopf algebras of finite type 3.

H = k[X

, . . . , X

, Y

, Y

, . . . , Y

, Y

1

] ' k[X ]

⊗ k[Y, Y

]

Primitive elements: k -space kX

+ · · · + kX

, dimension d

.

Group-like elements: multiplicative group hY

, . . . , Y

i, rank d

.

G = G

× G

k [G] = H, G(K ) = Hom (H, K ).

Bicommutative Hopf algebras of finite type 3.

H = k[X

, . . . , X

, Y

, Y

, . . . , Y

, Y

1

] ' k[X ]

⊗ k[Y, Y

]

The category of commutative linear algebraic groups over k G = G

× G

is anti-equivalent to the category of Hopf algebras of finite type which are bicommutative (commutative and cocomutative)

H = k [G].

The vector space of primitive elements has dimension d

while

the rank of the group-like elements is d

.

Other examples

If W is a k-vector space of dimension `

, Sym(W ) is a bicommutative Hopf algebra of finite type, anti-isomorphic to k [G

]:

For a basis ∂

, . . . , ∂

of W , Sym(W ) ' k[∂

, . . . , ∂

].

Other examples

If W is a k-vector space of dimension `

, Sym(W ) is a bicommutative Hopf algebra of finite type, anti-isomorphic to k [G

].

If Γ is a torsion free finitely generated Z-module of rank `

, then the group algebra k Γ is again a bicommutative Hopf algebra of finite type, anti-isomorphic to k[G

]:

For a basis γ

, . . . , γ

of Γ, kΓ ' k [γ

, γ

, . . . , γ

, γ

1

].

Other examples

If W is a k-vector space of dimension `

, Sym(W ) is a bicommutative Hopf algebra of finite type, anti-isomorphic to k [G

].

If Γ is a torsion free finitely generated Z-module of rank `

, then the group algebra k Γ is again a bicommutative Hopf algebra of finite type, anti-isomorphic to k[G

].

The category of bicommutative Hopf algebras of finite type is equivalent to the category of pairs (W, Γ) where W is a k -vector space and Γ is a finitely generated Z-module:

H = Sym(W ) ⊗ kΓ.

Commutative linear algebraic groups over Q G = G

× G

d = d

+ d

G(Q) = Q

× (Q

)

(β

, . . . , β

, α

, . . . , α

)

Commutative linear algebraic groups over Q G = G

× G

d = d

+ d

G(Q) = Q

× (Q

)

exp

: T

(G) = C

−→ G(C) = C

× (C

)

(z

, . . . , z

) 7−→ (z

, . . . , z

, e

, . . . , e

)

Commutative linear algebraic groups over Q G = G

× G

d = d

+ d

G(Q) = Q

× (Q

)

exp

: T

(G) = C

−→ G(C) = C

× (C

)

(z

, . . . , z

) 7−→ (z

, . . . , z

, e

, . . . , e

) For α

and β

in Q,

exp

(β

, . . . , β

, log α

, . . . , log α

) ∈ G(Q)

Baker’s Theorem. If

β

+ β

log α

+ · · · + β

log α

= 0 with algebraic β

and α

, then

1. β

= 0

2. If (β

, . . . , β

) 6= (0, . . . , 0), then log α

, . . . , log α

are Q- linearly dependent.

3. If (log α

, . . . , log α

) 6= (0, . . . , 0), then β

, . . . , β

are Q-

linearly dependent.

Example: (3 − 2 √

5) log 3 + √

5 log 9 − log 27 = 0.

Example: (3 − 2 √

5) log 3 + √

5 log 9 − log 27 = 0.

Corollaries.

1. Hermite-Lindemann (n = 1): transcendence of e, π, log 2, e

√2

.

Example: (3 − 2 √

5) log 3 + √

5 log 9 − log 27 = 0.

Corollaries.

1. Hermite-Lindemann (n = 1): transcendence of e, π, log 2, e

√2

.

2. Gel’fond-Schneider (n = 2, β

= 0): transcendence of 2

√2

, log 2/ log 3, e

.

Example: (3 − 2 √

5) log 3 + √

5 log 9 − log 27 = 0.

Corollaries.

1. Hermite-Lindemann (n = 1): transcendence of e, π, log 2, e

√2

.

2. Gel’fond-Schneider (n = 2, β

= 0): transcendence of 2

√2

, log 2/ log 3, e

.

3. Example with n = 2, β

6= 0: transcendence of Z

dx = 1

log 2 + π

√ ·

Values of exponential polynomials Proof of Baker’s Theorem. Assume

β

+ β

log α

+ · · · + β

log α

= log α

(B

) (Gel’fond–Baker’s Method)

Functions: z

, e

, . . . , e

, e

Points: Z(1, log α

, . . . , log α

) ∈ C

Derivatives: ∂/∂z

, (0 ≤ i ≤ n − 1).

Values of exponential polynomials Proof of Baker’s Theorem. Assume

β

+ β

log α

+ · · · + β

log α

= log α

(B

) (Gel’fond–Baker’s Method)

Functions: z

, e

, . . . , e

, e

Points: Z(1, log α

, . . . , log α

) ∈ C

Derivatives: ∂/∂z

, (0 ≤ i ≤ n − 1).

n + 1 functions, n variables, 1 point, n derivatives

Another proof of Baker’s Theorem. Assume again β

+ β

log α

+ · · · + β

log α

= log α

(B

) (Generalization of Schneider’s method)

Functions: z

, z

, . . . , z

, e

α

· · · α

=

exp{z

+ z

log α

+ · · · + z

log α

}

Points: {0} × Z

+ Z(β

, , . . . , β

) ∈ C

Derivative: ∂/∂z

.

Another proof of Baker’s Theorem. Assume again β

+ β

log α

+ · · · + β

log α

= log α

(B

) (Generalization of Schneider’s method)

Functions: z

, z

, . . . , z

, e

α

· · · α

=

exp{z

+ z

log α

+ · · · + z

log α

} Points: {0} × Z

+ Z(β

, , . . . , β

) ∈ C

Derivative: ∂/∂z

.

Six Exponentials Theorem. If x

, x

are two complex numbers which are Q-linearly independent and if y

, y

, y

are three complex numbers which are Q-linearly independent, then one at least of the six numbers

e

(i = 1, 2, j = 1, 2, 3)

is transcendental.

Proof of the six exponentials Theorem

Assume x

, . . . , x

are Q-linearly independent numbers and y

, . . . , y

are Q-linearly independent numbers such that

e

∈ Q for i = 1, . . . , a, , j = 1, . . . , b with ab > a + b.

Functions: e

(1 ≤ i ≤ a) Points: y

∈ C (1 ≤ j ≤ b)

a functions, 1 variable, b points, 0 derivative

Linear Subgroup Theorem G = G

× G

, d = d

+ d

.

W ⊂ T

(G) a C-subspace which is rational over Q. Let `

be its dimension.

Y ⊂ T

(G) a finitely generated subgroup with Γ = exp(Y ) contained in G(Q) = Q

× (Q

)

. Let `

be the Z-rank of Γ.

V ⊂ T

(G) a C-subspace containing both W and Y . Let n be the dimension of V .

Hypothesis:

n(`

+ d

) < `

d

+ `

d

+ `

d

n(`

+ d

) < `

d

+ `

d

+ `

d

d

+ d

is the number of functions

d

are linear

d

are exponential

n is the number of variables

`

is the number of derivatives

`

is the number of points

d

d

`

`

n

Baker B

1 n n 1 n

Baker B

n 1 1 n n

Six exponentials 0 a 0 b 1

d

d

`

`

n

Baker B

1 n n 1 n

Baker B

n 1 1 n n

Six exponentials 0 a 0 b 1 Baker:

n(`

+ d

) = n

+ n

`

d

+ `

d

+ `

d

= n

+ n + 1

Six exponentials: a + b < ab

n(`

+ d

) = a + b

duality:

(d

, d

, `

, `

) ←→ (`

, `

, d

, d

)

d dz

z

e

z=y

=

d dz

z

e

z=x

.

Fourier-Borel duality:

(d

, d

, `

, `

) ←→ (`

, `

, d

, d

)

d dz

z

e

z=y

=

d dz

z

e

z=x

. L

: f 7−→

d dz

f (y ).

f

(z ) = e

, L

(f

) = ζ

e

. L

(z

f

) =

d dζ

L

(f

).

For v = (v

, . . . , v

) ∈ C

, set D

= v

∂

∂z

+ · · · + v

∂

∂z

·

Let w

, . . . , w

0

, u

, . . . , u

0

, x and y in C

, t ∈ N

and s ∈ N

. For z ∈ C

, write

(uz )

= (u

z )

· · · (u

0

z )

and D

= D

1

· · · D

0

. Then

D

(uz )

e

= D

(wz )

e

Interpretation of the duality in terms of Hopf algebras following St´ ephane Fischler

Let C

be the category with

objects: (G, W, Γ) where G = G

× G

, W ⊂ T

(G) is rational over Q and Γ ∈ G(Q) is finitely generated

morphisms: f : (G

, W

, Γ

) → (G

, W

, Γ

) where f : G

→ G

is a morphism of algebraic groups such that f (Γ

) ⊂ Γ

and f induces a morphism

df : T

(G

) −→ T

(G

)

Let H be a bicommutative Hopf algebra over Q of finite type.

Denote by d

the dimension of the Q-vector space of primitive elements and by d

the rank of the group of group-like elements.

Let H

be also a bicommutative Hopf algebra over Q of finite type, `

the dimension of the space of primitive elements and `

the rank of the group-like elements.

Let h·i : H × H

−→ Q be a bilinear product such that

hx, yy

i = h∆x, y ⊗ y

i and hxx

, yi = hx ⊗ x

, ∆y i.

Let C

be the category with

objects: (H, H

, h·i) pair of Hopf algebras with a bilinear product as above.

morphisms: (f, g ) : (H

, H

, h·i

) → (H

, H

, h·i

) where f : H

→ H

and g : H

→ H

are Hopf algebras morphisms such that

hx

, g (x

)i

= hf (x

), x

i

.

St´ ephane Fischler: The categories C

and C

are equivalent.

Further, Fourier-Borel duality amounts to permute H and H

.

Consequence: interpolation lemmas are equivalent to zero

estimates.

St´ ephane Fischler: The categories C

and C

are equivalent.

For R ∈ C[G], ∂

, . . . , ∂

∈ W and γ ∈ Γ, set

hR, γ ⊗ ∂

· . . . · ∂

i = ∂

· . . . · ∂

R(γ ).

Conversely, for H

= C[G] and H

= Sym(W ) ⊗ k Γ, consider

Γ −→ G(C)

γ 7−→ R 7→ hR, γ i

and W −→ T

(G)

∂ 7−→ R 7→ hR, ∂ i

St´ ephane Fischler: The categories C

and C

are equivalent.

Further, Fourier-Borel duality amounts to permute H and H

. Open Problems:

• Define n associated with (G, Γ, W ) in terms of (H, H

, h·i)

St´ ephane Fischler: The categories C

and C

are equivalent.

Further, Fourier-Borel duality amounts to permute H and H

. Open Problems:

• Define n associated with (G, Γ, W ) in terms of (H, H

, h·i)

• Extend to non linear commutative algebraic groups (elliptic

curves, abelian varieties, and generally semi-abelian varieties)

St´ ephane Fischler: The categories C

and C

are equivalent.

Further, Fourier-Borel duality amounts to permute H and H

. Open Problems:

• Define n associated with (G, Γ, W ) in terms of (H, H

, h·i)

• Extend to non linear commutative algebraic groups (elliptic curves, abelian varieties, and generally semi-abelian varieties)

• Extend to non bicommutative Hopf algebras (of finite type to

start with)

St´ ephane Fischler: The categories C

and C

are equivalent.

Further, Fourier-Borel duality amounts to permute H and H

. Open Problems:

• Define n associated with (G, Γ, W ) in terms of (H, H

, h·i)

• Extend to non linear commutative algebraic groups (elliptic curves, abelian varieties, and generally semi-abelian varieties)

• Extend to non bicommutative Hopf algebras (of finite type to start with)

• (?) Transcendence results on non commutative algebraic groups