Poizat’s bad fields and quantum groups

Boris Zilber University of Oxford 27 September 2004 http://www.maths.ox.ac.uk/∼zilber

Bad fields

Definition A bad field is a structure (K, +, ·, G) with MR(K) = N > 1 and G < K^{×}, a multiplicative subgroup, MR(G) = 1.

Problem (197?) Do bad field exist?

Theorem (Baldwin and Holland, 2002) Yes, for each N > 1, if we drop the requirement that G is a group.

Theorem ( B.Poizat, 2000) There exist an almost bad fields (K, +, ·, G) with MR(K) = ω × N and G < K^{×}, a multiplicative subgroup, MR(G) = ω.

Proof. Based on Hrushovski’s construction.

Easy case: G = P is just a subset (of black points).

Predimension: δ(X) = N ·tr.deg(X)−size(X ∩P).

Axioms: GSCH: For distinct x_{1}, …, x_{k} ∈ P,

N · tr.deg(x_{1}, …, x_{k}) − k ≥ 0. EC: Let V ⊆ ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C) which has a point ⟨a_{1}, …, a_{k}⟩ ∈ V satisfying: a_{i} ≠ a_{j} and a_{i} ∉ acl(C), i ≠ j, i ≤ k (P -free) and

tr.deg(a_{i_{1}}, …, a_{i_{m}}) ≥ \frac{m}{N}, for any i_{1} < ⋯ < i_{m} ≤ k

(P -normal).

Then there is ⟨a_{1}, …, a_{k}⟩ ∈ V ∩ P^{k}.

Claim. GSCH and EC for (K, +, ·, P) are first order. Proof. Easy □ Difficult case: G is a subgroup of the multiplicative group (green points): Predimension

δ(X) = N · tr.deg(X) − mult.rk(X ∩ G). Axioms:

GSCH: For multiplicatively independent x_{1}, …, x_{k} ∈ G,

N · tr.deg(x_{1}, …, x_{k}) − k ≥ 0.

EC: Let V ⊆ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C) and which is G-free and G-normal. Then there is ⟨a_{1}, …, a_{k}⟩ ∈ V ∩ G^{k}.

Claim. GSCH and EC for (K, +, ·, G) are first order. Proof. Uses Ax’s Theorem on Schanuel’s conjecture for function fields. □

Problem. Explain these examples analytically.

Solutions, for N = 2. K = ℂ. Case: P is a “generic” subset of ℂ. Let ϵ, α be algebraic numbers, ϵ ∉ ℝ ∪ iℝ, α ∉ ℚ, αℝ ≠ ϵℝ. Let f be a Liouville function (A.Wilkie) and

P = {f(ϵt + αq) : t ∈ ℝ, q ∈ ℚ}

Theorem (ℂ, +, ·, P) is a model of Poizat’s “black points” theory.

Case: G is a multiplicative subgroup of ℂ.

Let

G = {exp(ϵt + αq) : t ∈ ℝ, q ∈ ℚ}

Theorem. Assume Schanuel’s conjecture. Then (ℂ, +, ·, G) is a model of Poizat’s “green points” theory.

Problem 1. Do the general case N. Remark Necessarily, the real dimension of the bad subset (P or G) must be \frac{2}{N} . Thus, the theory of fractional dimensions needs to be involved in the construction.

Let h ∈ ℝ ∖ ℚ, ϵ ∈ (ℂ ∖ ℝ ∪ iℝ),

G = {exp(ϵt + 2πihm) : t ∈ ℝ, m ∈ ℤ}

Γ = {exp(2πihm) : m ∈ ℤ}

Theorem Assume Schanuel’s conjecture. Then 1.Th(ℂ, +, ·, G, Γ) is superstable,

U(ℂ) = ω · 2, U(G) = ω, U(Γ) = 1.

2. The Miller-Speissegger spiral G^{0} = exp(ϵℝ) is type-definable in (ℂ, +, ·, G) as the connected component of G.

3. The field of reals is L_{ω_{1},ω}-definable in (ℂ, +, ·, G).

Problems 2.Describe canonical models in the elementary class Th(ℂ, +, ·, G, Γ).

3.Put (ℂ, +, ·, G, Γ) in the context of analytic Zariski structures.

The quantum torus

Theorem Given an algebraically closed field F and its cyclic multiplicative subgroup Γ = ⟨q⟩, the two-sorted structure

(F, +, ·) ⟶^{θ} (T, ·)

(Dom θ = F^{∗}, θ a homomorphism, ker θ = Γ) has an analytic Zariski structure on both sorts F and T.

Consider the F-vector space H = H(T) of local functions on infinitesimal neighborhood V of 1 ∈ T :

H(T) = {ψ : V ⊆^{∗}T → ^{∗}F}.

Example The inverse x = θ^{−1} to the map θ : ^{∗}F → ^{∗}T is well-defined on V. Consequently, for every k ∈ ℤ

x^{k} : ^{∗}T → ^{∗}F

is well-defined. So, x^{k} ∈ H(T), k ∈ ℤ.

There is an algebra A(T) of definable linear operators acting on H(T) :

U : x^{k} → x^{k+1},
V : x^{k} → q^{k}x^{k}.

A(T) := ⟨V, U : V U = qUV ⟩.

Theorem Assuming Schanuel’s conjecture the 3sorted structure below is superstable and is known to have some analytic Zariski properties

\begin{matrix}(ℂ, +, h·) & ⟶^{exp v} & (ℂ, +, ·) \\ ↓_{exp h^{−1}v} & ↓ & θ \\ (ℂ, +, ·) & ⟶ & T = ℂ^{∗}/Γ \end{matrix}

Γ = ⟨q⟩, q = exp(2πih).

In this language H(T) contains also the well-defined local functions

x^{kh} : ^{∗}T → ^{∗}F

with the action of the operators

U : x^{kh} → q^{k}x^{kh},
V : x^{kh} → x^{(k+1)h}.

Using the obvious symmetry between U and V we obtain the formal correspondence between the eigenvectors of the operators U and V :

x^{kh} ∼ ∑_{m∈ℤ} q^{−km}x^{m}.

Problem 4. Give a meaning to this formula.

Problem 5. Add the bad subgroup

G = exp(ϵℝ + 2πihℤ)

to the language and include the reals into the picture.

The quantum SL_{q}(2, ℂ)

Consider the action of the group

ℤ × ℤ ≅ Γ × Γ = {(q^{m}, q^{n}) : m, n ∈ ℤ} on SL(2, ℂ) :

( \begin{matrix}X Y \\ Z V \end{matrix} ) ⟶^{(m,n)} ( \begin{matrix}Xq^{n}  Y \\ Zq^{m} \frac{XV +Y Z(1−q^{m})}{Xq^{n}} \end{matrix} )

This gives rise to the space of orbits

SL_{q}(2, ℂ) = Γ × Γ\SL(2, ℂ).

Theorem The two-sorted structure

SL(2, ℂ) ⟶^{θ} SL_{q}(2, ℂ)

is superstable and analytic Zariski in both sorts.

The above mentioned method of constructing an algebra A of linear operators acting on the space H of local functions

∗SL_{q}(2, ℂ) →^{∗} SL(2, ℂ)

produces the ℂ-algebra with generators a, b, c, d and defining relations

ab = qba
bd = qdb
ac = qca
cd = qdc
bc = cb
ad − da = (q − q^{−1})bc
ad − qbc = da − q^{−1}bc = 1.

This can be naturally made a Hopf algebra (with a comultiplication and a counit). This Hopf algebra

O_{q}(SL_{2}(ℂ))

is by definition the (algebraic) quantum SL(2, ℂ).

Problem 6. Consider SL_{q}(2, ℂ) in an expanded language involving the reals (as in Problem 5).

Problem 7. Study the model theory of the quantum unitary group U_{q}(2, ℂ) and the quantum orthogonal group O_{q}(3). Look for ’bad’ stable groups related to these structures.