Zariski structures and noncommutative geometry

B. Zilber

University of Oxford

http://www.people.maths.ox.ac.uk/ ∼zilber: Zariki Geometries (forthcoming book); A class of quantum Zariski geometries;

Non-commutative Zariski geometries and their classical limit; Quantum Harmonic Oscillator as a Zariski Geometry.

Zariski structures

Zariski structures (1993, E.Hrushovski and B.Zilber) are on the very top of the (logical) stability hierarchy. The ones for which a fine classification theory is possible.

Let M be a structure given with a family of basic relations (subsets of M^{n}) called Zariski closed.

We postulate for a Zariski structure M:

Let M be a structure given with a family of basic relations (subsets of M^{n}) called Zariski closed.

We postulate for a Zariski structure M:

(T) Zariski closed sets form a Noetherian Topology on M^{n}, all n.

(P) Projection pr(S) ⊆ M^{n} of a closed set S ⊆ M^{n+1} is constructible (= Boolean combination of closed).

(D) Dimension dim S to any closed S ⊆ M^{n} is assigned.

(AF) Addition formula: dim S = dim pr(S) + min_{a∈pr(S)} dim(pr^{−1}(a) ∩ S)

for any closed irreducible S.

(FC) Fiber condition: for each k, the set

{a ∈ M^{n−1} : dim(S ∩ pr^{−1}(a)) > k} is constructible.

(PS) Pre-smoothness: For any closed irreducible S_{1}, S_{2} ⊆ M^{n},

dim S_{1} ∩ S_{2} ≥ dim S_{1} + dim S_{2} − dim M^{n} in each component.

Known classes of Zariski structures 1. Algebraic varieties over an algebraically closed field with respect to the usual Zariski topology.

Known classes of Zariski structures

1. Algebraic varieties over an algebraically closed field with respect to the usual Zariski topology.

2. Compact complex spaces, Zariski closed subset of M^{n} ≡ analytic subset.

2. Compact complex spaces, Zariski closed subset of M^{n} ≡ analytic subset.

3. Proper analytic varieties over complete algebraically closed non-Archimedean valued fields (rigid analytic geometry) .

2. Compact complex spaces, Zariski closed subset of M^{n} ≡ analytic subset.

3. Proper analytic varieties over complete algebraically closed non-Archimedean valued fields (rigid analytic geometry).

4. A large class of non-commutative geometries About the term Geometry. Geometric tradition explains ”spaces” as given locally by co-ordinate functions (into ℝ or ℂ). This follows the physicist’s paradigm that the ultimate data is given in numbers.

Classification Theorem (Hrushovski, Zilber 1993) For any non-linear Zariski geometry M there is an algebraically closed field 𝔽 and a nonconstant meromorphic function

f : M → 𝔽.

In particular, if dim M = 1 then there is a smooth projective algebraic curve C_{M} and a Zariski-continuous finite covering map

p : M → C_{M}(𝔽),

the image of any relation on M is just an algebraic relation on C_{M}.

[figure: M; ^{C^{M}}; p; Zariski structure and its algebraic projection]

[figure: M; ^{C^{M}}; p; Zariski structure and its algebraic projection; 𝔽; Field of numbers]

[figure: M; ^{C^{M}}; p; Zariski structure and its algebraic projection; 𝔽; Field of numbers]

[figure: M; ^{C^{M}}; p; Zariski structure and its algebraic projection; 𝔽; Field of numbers]

[figure: M; ^{C^{M}}; p; Zariski structure and its algebraic projection; 𝔽; Field of numbers; Regular coordinate functions from M; detect the classical part C_{M} only]

𝔽[M] = {f : M → 𝔽 regular } 𝔽[M] = 𝔽(C_{M}), C_{M} = M/E,

E an equivalence relation on M. In general there may be Zariski-continuous “entangling” arrows (action)

γ : M → M, γ ∈ Γ

which make E non-splitting.

The initial example: 2-cover of the affine line.

Classification theorem revisited. How to recover the hidden relations in terms of “coordinate functions”?

Classification theorem revisited. How to recover the hidden relations in terms of “coordinate functions”? Extend the 𝔽-algebra of definable functions 𝔽[M] to the 𝔽-space of semi-definable functions H[M]. Every Zariski bijection γ generates an 𝔽-linear transformation of H[M] :

U_{γ} : f ↦ f^{γ}
f ∈ H[M], f^{γ}(x) = f(γx).

Also, any y ∈ 𝔽[M] gives rise to an 𝔽-linear Y : f ↦ y · f Classification theorem revisited. How to recover the hidden relations in terms of “coordinate functions”? Extend the 𝔽-algebra of definable functions 𝔽[M] to the 𝔽-space of semi-definable functions H[M]. Every Zariski bijection γ generates an 𝔽-linear transformation of H[M] :

U_{γ} : f ↦ f^{γ}

Also, any y ∈ 𝔽[M] gives rise to an 𝔽-linear Y : f ↦ y · f The operator algebra A[M] generated by all the U_{γ} and Y ’s contains data sufficient to recover M.

M \overline{}𝔽[M]

M \overline{}𝔽[M] ⊂ H[M]

[figure: M; 𝔽[M] ⊂ H[M]; A[M]]

[figure: M; 𝔽[M] ⊂ H[M]; A[M]]

Remarks

1. Elements of H[M] are not uniquely definable within M, so should be considered as auxiliary, not well-defined.

Remarks

1. Elements of H[M] are not uniquely definable within M, so should be considered as auxiliary, not well-defined.

2. A[M] and its elements are uniquely defined (up to the choice of the language) so can be seen as observables.

\{ \begin{matrix}universe \\ of M \end{matrix} \} ⟷ \{ \begin{matrix}class of 1-dim rep of \\ a commutative B ≤ A[M] \end{matrix} \}

Auxiliary functions from H[M] induce a formal C^{∗}- algebra structure on A[M], with a notion of adjointness and a meaning of a positive eigenvalue.

B is generated by self-adjoint operators and M consists of positive eigenspaces (so elements of M may be called states).

The operators U_{γ} ∈ A[M] become unitary and act on B by conjugation. This corresponds to the action of the γ on M.

A(M) for the ℓ-cover of the affine line (ϵ ∈ 𝔽, ϵ^{ℓ} = 1):

HY = YH; HZ = ZH; YZ = ZY; Y^{ℓ} = I; Z^{ℓ} = I; UH − HU = hU; VH − HV = ihV; UY = ϵYU; YV = VY; ZU = UZ; VZ = YZV; E = U^{−1}V^{−1}UV; E^{ℓ} = I; UE = EU; VE = EV.

Y, Z, U, V and E unitary, H self-adjoint (slightly simplified).

Inverse problem. Start with a noncommutative algebra A and produce a Zariski M = M[A].

Quantum algebras at roots of unity

We assume for a “quantum algebra A at roots of unity”:

1. A is an affine unital 𝔽-algebra, finite-dimensional over its centre Z(A). 𝔽 algebraically closed.

2. Isomorphism classes of generic irreducible Amodules are in a bijective correspondence with an open subset V^{0} ⊆ Max Z(A) of the affine variety.

3. Generic irreducible modules allow a uniform choice of canonical bases degenerating regularly outside V^{0} preserving the dimension.

Examples (for q^{ℓ} = 1)

1. A = ⟨U, V : UV = qV U⟩ Manin’s quantum plane

2. A = U_{q}(𝔰𝔩_{2}) quantised U_{q}(𝔰𝔩_{2}) as a Hopf algebra (quantum group)

3. A = O_{q}(SL_{2}) quantised co-ordinate Hopf algebra of SL_{2} (quantum group) ...

We associate with every such A the bundle

mod_{A}^{(ℓ)} = {𝔪_{a} : a ∈ Max Z(A)}

of ℓ-dimensional A-modules 𝔪_{a} (with or without selected canonical bases) over the algebraic variety V_{A} = Max Z(A).

For each X ∈ A the action of X on 𝔪_{a} is a part of the structure and is given uniformly in a.

mod_{A}^{(ℓ)} = {𝔪_{a} : a ∈ Max Z(A)}

of ℓ-dimensional A-modules 𝔪_{a} (with or without selected canonical bases) over the algebraic variety V_{A} = Max Z(A).

For each X ∈ A the action of X on 𝔪_{a} is a part of the structure and is given uniformly in a.

Typically, other finite-dimensional A-modules as well as morphism maps between modules are definable in mod_{A}^{(ℓ)}, so we expect that up to interdefinability

the structure mod_{A}^{(ℓ)} is equivalent to mod_{A}, the category of finite dimensional A-modules.

mod_{A}^{(ℓ)} = {𝔪_{a} : a ∈ Max Z(A)}

of ℓ-dimensional A-modules 𝔪_{a} (with or without selected canonical bases) over the algebraic variety V_{A} = Max Z(A).

For each X ∈ A the action of X on 𝔪_{a} is a part of the structure and is given uniformly in a.

One may also consider the infinite-dimensional Amodule

H := ∑_{a} 𝔪_{a}

with or without a choice of canonical bases in each 𝔪_{a}.

Theorem The structure mod_{A}^{(ℓ)} is a Zariski geometry, with respect to a Zariski topology.

The 𝔽-algebra A is determined by mod_{A}^{(ℓ)} as the algebra of definable linear transformations of H (equivalently, of the vector bundle 𝔪_{a}).

mod_{A}^{(ℓ)} is not definable in commutative algebraic geometry, in general.

For A commutative, mod_{A}^{(ℓ)} is the trivial line bundle over Max A, and so the geometry is equivalent to that of the algebraic variety Max A, .

Remark mod_{A}^{(ℓ)} is not the unique construction satisfying the properties above. Other constructions produce definably equivalent Zariski geometries. In all the cases (known to us) these are equivalent to mod_{A}, the category of finite dimensional A-modules.

Not a root of unity case.

Not a root of unity case. The quantum harmonic oscillator

The quantum harmonic oscillator

A = ⟨P, Q : PQ − QP = ih⟩

as C^{∗}-algebra. In mathematical physics

H = \frac{1}{2}(P^{2} + Q^{2}),

the Hamiltonian of the harmonic oscillator.

P, Q and H are self-adjoint.

A = ⟨P, Q : PQ − QP = ih⟩

as C^{∗}-algebra. In mathematical physics

H = \frac{1}{2}(P^{2} + Q^{2}),

the Hamiltonian of the harmonic oscillator.

C_{+} = \frac{1}{2}(P + iQ), C_{−} = \frac{1}{2}(P − iQ)

the creation and annihilation operators;

C_{+}C_{−} = H+\frac{h}{2}, C_{−}C_{+} = H−\frac{h}{2}, C_{+}C_{−}−C_{−}C_{+} = h.

[figure: The universe E = {±e_{x} : x ∈ ℂ}; +e_{x}; −e_{x}; −e_{y}; +e_{y}]

[figure: The universe E = {±e_{x} : x ∈ ℂ}; +e_{x}; −e_{x}; −e_{y}; +e_{y}; ℂ; x; y]

[figure: The universe E = {±e_{x} : x ∈ ℂ}; Think of line bundle; spanned by e_{x} over ℂ; ℂ; V_{x}; V_{z}]

[figure: The universe E = \{±e_{x} : x ∈ ℂ\}; Think of line bundle; spanned by e_{x} over ℂ; ℂ; V_{x}; V_{z}; H + \frac{1}{2} defines the linear maps; V_{x} → V_{x}; v ↦ x v]

[figure: The universe E = {±e_{x} : x ∈ ℂ}; Think of line bundle; spanned by e_{x} over ℂ; ℂ; V_{x}; V_{z}; C_{+} and C_{−} define linear maps; C_{+} : V_{x} → V_{x+h}; C_{−} : V_{x} → V_{x−h}]

[figure: C_{+} and C_{−} define linear maps; C_{+} : V_{x} → V_{x+h}; λe_{x} ↦ \sqrt{x}λe_{x+h}; C_{−} : V_{x} → V_{x−h}; λe_{x} ↦ \sqrt{x − h}λe_{x−h}; C_{+}; x; x + h; xe_{x}; C_{−}; e_{x}; \sqrt{x}e_{x+h}; −e_{x}; C_{+}; −\sqrt{x}e_{x+h}]

Theorem The structure E(A) corresponding to the quantum harmonic oscillator is a 1-dimensional (complex) Zariski geometry.

Theorem The structure E(A) corresponding to the quantum harmonic oscillator is a (complex) Zariski geometry.

When one applies the full restrictions imposed by the C^{∗}-algebra structure one gets the real part of E(A), which is discrete in this case.

[figure: 2h; 3h; 4h; h; 0]

Problem Explain model-theoretically transitions between bases of H-eigenvectors, P-eigenvectors and Q-eigenvectors.

Problem Explain model-theoretically transitions between bases in H-eigenvectors, P-eigenvectors and Q-eigenvectors.

Over the field of characteristic p the algebra A is a “quantum algebra at roots of unity” and so M(A) is a Zariski geometry again.

Problems and projects

1. Establish a right category of geometric objects corresponding to non-commutative algebras A :

- as “algebro-geometric” coordinate algebras,

- as C^{∗}-algebras,

- understand the interplay of the algebro-geometric and real geometric structures.

2. Develop a deformation (approximation) theory at the level of geometric objects

- e.g. as h → 0

- as a root of unity converges to a generic q

- to explain how (and if) an elliptic curve deforms into a quantum torus.

2. Develop a deformation (approximation) theory at the level of geometric objects

- e.g. as h → 0

- as a root of unity converges to a generic q

- to explain how (and if) an elliptic curve deforms into a quantum torus.

3. Explain model-theoretically the meaning of various non-convergent sums of maths physics.

Example.

Theorem There is a well-defined Gromov-Hausdorff limit of the ℓ-cover of the affine line

lim_{\frac{1}{ℓ}=h→0} M_{h} = \{ \begin{matrix}real differentiable manifold = \\ U(1)-gauge field over a 2-dim real manifold \end{matrix} \}

The limit of the unitary operators U and V correspond to covariant differentiation on the gauge field. For the ℓ-cover of the torus ℂ^{∗}, the connection of the gauge field is of non-constant curvature.