数理解析研究所講究録
第 1794 巻 2012 年 73-81

73First-order theory of quantum 2-tori
Masanori Itai (板井$T_{q}^{2}$昌典)

Department of Mathematical Sciences(東海大学理学部情報数理学科)

Tokai University, Hiratsuka, Japan

Abstract

We discuss the first-order description of non-commutative quantum 2-tori.

1

Introduction

Quantum tori are geometric objects associated with non-commutativealgebras
with generating multiplicative groups. When is a rootof unity, we have a quantum torus which is a Zariski structure (Zilberresult).
On the other hand, when generates a cyclic group of countableorder we hope to construct new kinds of analytic Zariski structures as
non-commutative quantum 2-tori.
In this note we give the first-order description of the quantum
2-tori given in [Zl].
$\mathcal{A}_{q}$

$q$

$q$

$s$

$q$

2

Description of the torus$T_{q}^{2}(\mathbb{C})$First we give the description of a quantum 2-torus defined over thecomplex numbers . Consider a C-algebra
generated by operators
satisfying$\mathbb{C}$$\mathcal{A}_{q}^{2}$$U,$ $U^{-1},$ $V,$ $V^{-1}$$VU=qUV$,$UU^{-1}=U^{-1}U=VV^{-1}=V^{-1}V=I$

74
where $q=e^{2\pi ih}$ withof$\mathbb{C}^{*}$$h\in \mathbb{R}$. Let$\Gamma_{q}=q^{Z}$

be a multiplicative subgroup.

itself is called a quantum torus.
In the literature, the C-algebra
over associIn this paper, however, the quantum 2-torus
is the 3-sorted structure
and the groupated with the algebrawith the actions and satisfying$\mathcal{A}_{q}^{2}$$T_{q}^{2}(\mathbb{C})$$\Gamma_{q}$$\mathcal{A}_{q}^{2}$

:
:

$V$

and$V$where :, andWe also have$\mathbb{C}^{*}/\Gammaarrow \mathbb{C}^{*}$$u(\gamma u, v)\mapsto\gamma uu(\gamma u, v)$(1)$u(\gamma u, v)\mapsto vu(q^{-1}\gamma u, v)$

:
:

$U$

$\phi$$V$$U$$(U_{\phi}, V_{\phi}, \mathbb{C}^{*})$$U$$\mathbb{C}$$v(\gamma v,u)\mapsto uv(q\gamma v, u)$(2)$v(\gamma v, u)\mapsto\gamma vv(\gamma v, u)$is a (non-definable) “ choice fUnction”, $u(\gamma u, v)\in$.

$v(\gamma v, u)\in V_{\phi}$$U_{\phi}$$U^{-1}V^{-1}$

and$U^{-1}$:::$V^{-1}$$u(\gamma u, v)\mapsto\gamma^{-1}u^{-1}u(\gamma u, v)$(3)$u(\gamma u, v)\mapsto v^{-1}u(q\gamma u, v)$$v(\gamma v, u)\mapsto u^{-1}v(q^{-1}\gamma v_{:}u)$(4)$v(\gamma v, u)\mapsto\gamma^{-1}v^{-1}v(\gamma v, u)$

There is also a function
plays as an inner product$\{\cdot|\cdot\}$

called the pairing function which(5)$\langle\cdot|\cdot\}:(V_{\phi}\cross U_{\phi})\cup(U_{\phi}\cross V_{\phi})arrow\Gamma$

whose properties are given in the next subsection.

2.1
Fix$\langle u,$More details$\phi$:$\mathbb{C}^{*}/\Gammaarrow \mathbb{C}^{*}$$v\rangle\in\Phi^{2}$. Put $\Phi=$ ran$(\phi)$. We work with$\Phi^{2}$. Take. Let$U_{\langle u,v\rangle}$$;=$$\{\gamma_{1}\cdot u(\gamma_{2}u, v):\gamma_{1}.\gamma_{2}\in\Gamma\}$$V_{\langle u,v\rangle}$$;=$$\{\gamma_{1}\cdot v(\gamma_{2}v, u):\gamma_{1}.\gamma_{2}\in\Gamma\}$

Then let$U_{\phi}$$;=$$\bigcup_{\langle u,v\rangle\in\Phi^{2}}U_{\{u,v\rangle}$$V_{\phi}$$;=$$\bigcup_{\langle u,v\rangle\in\Phi^{2}}V_{\{u,v\rangle}$

,.

,.

(6)

(7)

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Consider the following sets$\mathbb{C}^{*}U_{\phi}$$\mathbb{C}^{*}V_{\phi}$$\mathbb{C}^{*}U_{\phi}$and$\mathbb{C}^{*}V_{\phi}$:$:=$$\{x\cdot u(\gamma u, v):\langle u, v\}\in\Phi^{2},$$:=$$\{x\cdot v(\gamma u, v):\langle u, v\rangle\in\Phi^{2}, x\in \mathbb{C}^{*}, \gamma\in\Gamma\}$$x\in \mathbb{C}^{*},$ $\gamma\in\Gamma\}$(8)

The pairing function (5) defined above satisfies the following quan, then for any $m,$
titative relations: takeandwe haveand$\langle u,$$k,$ $r,$ $s\in \mathbb{N}$$v\rangle\in\Phi^{2}$$q^{r}u(q^{k}u, v)\in U_{\{u,v\rangle}$$q^{s}v(q^{m}v, u)\in V_{\langle u,v\rangle}$(9)$\langle q^{s}v(q^{m}v, u)|q^{r}u(q^{k}u, \tau’)\}=q^{r-s}$and$\{q^{r}u(q^{k}u, v)|q^{s}v(q^{m}v, u)\rangle=q^{km+s-r}=\langle q^{s}v(q^{m}v, u)|q^{r}u(q^{k}u, v)\rangle^{-1}$.

(10)

For$v’\not\in\Gamma\cdot v$

or$u’\not\in\Gamma\cdot u$,$\langle q^{s}v(v’, u)|q^{r}u(u’, v)\}$

is not defined.
. The pairDefinition 1 Let be a choice function ofsatisfying (1) through (4)with operators
acting onandand the pairing function satisfying (9) and (10) is denoted $(U, V)_{\phi}$ .
$U,$ $V$

3$(U_{\phi}, V_{\phi})$$\mathbb{C}^{*}/\Gamma$$\phi$$V_{\phi}$$U_{\phi}$

The first-order theory of$T_{q}^{2}(F)$Proposition 4.4 of [Zl] claims that once $q=e^{2\pi ih}$ withisgiven any two structures of the form
are isomorphic. Following
Zilber argument in [Zl], we supply here details of his proof.
$h\in \mathbb{R}$$T_{q}^{2}(\mathbb{C})$$s$

Proposition 2 (Proposition 4.4, [Zl]) Given $q=e^{2\pi ih}$ with $h\in$any two structures of the form
are isomorphic over . In
other words, the isomorphism type ofdoes not depend on thesystem of representative .
$\mathbb{R}$$\mathbb{C}$$T_{q}^{2}(\mathbb{C})$$T_{q}^{2}(\mathbb{C})$$\Phi$, and set.
Proof: Fix a $q=e^{2\pi ih}$ with. Consider two strucLet
be two choice functions oftures $(U, V)_{\phi}$ and $(U, V)_{\psi}$ . We show that these two structures are
isomorphic.
Suppose picks
from theand picks
fromsame coset of.
$h\in \mathbb{R}$$\phi,$$\psi$$\Gamma=q^{Z}$$\mathbb{C}^{*}/\Gamma$$\phi$$\langle u_{g},$$\{u_{g},$$v_{g}\rangle$$v_{g}\rangle$$\mathbb{C}^{*}/\Gamma$$\psi$$\langle u_{0},$$v_{0}\}$

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andConsider the bases $\{u(q^{k}u_{g}, v_{g}) : k\in Z\}$ of$k\in Z\}$ of.
in the structure
are in the same coset ofandSince$U_{\langle u_{g},v_{g}\rangle}$$(U_{\phi}, V_{\phi})$$V_{\langle u_{g},v_{g}\rangle}$$\langle u_{0},$$s,$$\{v(q^{k}v_{g}, u_{g})$$\{u_{g},$$v_{0}\rangle$$\mathbb{C}^{*}/\Gamma$$v_{g}\rangle$

such that $u_{0}=q^{s}u_{g},$ $v_{0}=q^{t}v_{g}$ .
We want to transfer the structure ofthere are$t\in Z$

$U_{\angle u_{0},v_{0}\rangle}$$\bullet$$\bullet$$\bullet$$\bullet$and$V_{\langle u_{0},v_{\rangle}}$and$U_{\angle u_{g},v_{g}\}}$$V_{(u_{g},v_{g}\rangle}$toas follows. Set$u(u_{0}, v_{0});=q^{st}u(q^{s}u_{g}, v_{g})$,$u(q^{k}u_{0}, v_{0}):=v_{0}^{k}V^{-k}u(u_{0}, v_{0})$$v(v_{0}, u_{0}):=v(q^{t}v_{g}, u_{g})$,,$v(q^{k}v_{0}, u_{0}):=u_{0}^{-k}U^{k}v(v_{0}, u_{0})$,where $k\in$ Z. First notice that we have$\bullet$$u(q^{k}u_{0}, v_{0})$$=$$v_{0}^{k}V^{-k}u(u_{0}, v_{0})$$=$$(q^{t}v_{g})^{k}V^{-k}(q^{st}u(q^{s}u_{g}, v_{g}))$$=$$q^{kt}v_{g}^{k}q^{st}v^{-k}u(q^{s+k\prime}u_{g}, \prime v_{g})a_{+s}$,$q^{kt+st}u(q u_{g}, v_{g})$$=$$\bullet$$v(q^{k}v_{0}, u_{0})$$=$$u_{0}^{-k}U^{k}v(v_{0}, u_{0})$$q^{-ks}u^{-k}U^{k}v(g^{t}v_{g}, u_{g})$$q^{-ks}u^{\frac{g}{g}k}u_{9}^{k}v(g^{t+k}v_{g}, u_{g})$$q^{-sk}v(q^{k+t}v_{g}, u_{g})$.

These relations allow us to show that the operator
the set $U_{(u_{0},v_{0}\rangle}=\{u(q^{k}u_{0}, v_{0}) : k\in Z\}$ properly, i.e.
the rule (1) and (2).

andact onobey
and
$V$

$U$

$U$$(u-1)$$U(u(u_{0}, v_{0}))$$=$$U(q^{st}u(q^{s}ugc)g))$$=$$q^{st}U(u(q^{s}u_{g}, v_{g}))$$=$$q^{st}q^{s}u_{g}u(q^{s}u_{g}, v_{g})$$=$$u_{0}q^{st}u(q^{s}u_{g}, v_{g})$$=$$u_{0}u(u_{0}, v_{0})$,$(u-2)$$V(u(u_{0}, v_{0}))$$=$$V(q^{st}u(q^{s}u_{g}, v_{g}))$$q^{st}V(u(q^{s}u_{g}, v_{g}))$$q^{st}v_{g}u(q^{-1}q^{s}u_{9}, v_{g})$$q^{(s-1)t}q^{t}v_{g}u(q^{s-1}u_{g}, v_{g})$$q^{t}v_{g}q^{(s-1)t}u(q^{s-1}u_{g}, v_{g})$$v_{0}u(q^{s-1}u_{g}, q^{t}v_{g})$$v_{0}u(q^{-1}u_{0}, v_{0})$

,

$V$:

77$(u-3)$$U(u(q^{k}u_{0}, v_{0}))$$U(v_{0}^{k}V^{-k}u(u_{0}, v_{0}))$$=$$U(v_{0}^{k}V^{-k}q^{st}u(q^{s}u_{g}, v_{g}))$$U(q^{kt+st}u(q^{k+s}u_{g}, v_{g}))$$q^{kt+st}q^{k+s}u_{g}u(q^{k+s}u_{gg}’\iota))$$q^{k}u_{0}q^{kt+st}u(q^{k+s}u_{g}, v_{g})$$q^{k}u_{0}u(q^{k}u_{0}, v_{0})$,$(u-4)$$V(u(q^{k}u_{0}, v_{0}))$$V(q^{kt+st}u(q^{k+s}u_{g}, \prime u_{g}))$$=$$q^{kt+st}V(u(q^{k+s}u_{g}v_{g}))$$q^{kt+st}v_{g}u(q^{k+s-1}u_{g}, v_{g})$$q^{t}v_{g}q^{(k+s-1)t}u(q^{k+s-1}u_{g}, v_{g})$$v_{0}u(q^{k+s-1}u_{g}, v_{0})$$v_{0}u(q^{k-1}u_{0}, v_{0})$We also see that the operator$\{u(q^{k}v_{0}, u_{0}) : k\in Z\}$ properly, i.e.
as well.

$U$$U$

andand$V$$V$.

act on the set
obey the rule (1) and (2)$V_{\langle v_{0},u_{0}\rangle}=$$(v-1)$$U(v(v_{0}, u_{0}))$$=$$U(v(q^{t}v_{g}, u_{g}))$$=$$u_{g}v(q^{t+1}v_{g}, u_{g}))$$=$$u_{g}q^{s}q^{-s}v(q^{t+1}v_{g}, u_{g})$$=$$u_{0}v(qv_{0}, u_{0})$,$(v-2)$$V(v(v_{0}, u_{0}))$$=$$V(v(q^{t}v_{g}, u_{g}))$$=$$q^{t}v_{g}v(q^{t}v_{g}, u_{g}))$$=$$v_{0}v(v_{0}, u_{0})$,$(v-3)$$U(v(q^{k}v_{0}, u_{0}))$$=$$U(q^{-sk}v(q^{k+t}v_{g}, u_{g}))$$=$$q^{-sk}U(v(q^{k+t}v_{g}, u_{g}))$$=$$q^{-sk}u_{g}v(q^{k+t+1}v_{g}, u_{g})$$=$$q^{s}u_{g}q^{-s(k+1)}v(q^{k+1+t}v_{g}, u_{g})$$=$$u_{0}v(q^{k+1}v_{0}, u_{0})$,$(v-4)$$V(v(q^{k}v_{0}, u_{0}))$$=$$V(q^{-sk}v(q^{k+t}v_{g}, u_{g}))$$q^{-sk}V(v(q^{k+t}v_{g}u_{g}))$$=$$q^{-sk}q^{k+t}v_{g}v(q^{k+t}v_{g}, u_{g})$$q^{k+t}v_{g}q^{-sk}v(q^{k+t}v_{g}, u_{g})$$=$$q^{k}v_{0}v(q^{k}v_{0}, u_{0})$.

78
Finally the following relations tell us that we can properly transfer:the pairing function fromto$(U_{\langle u_{g},v_{g}\rangle}, V_{\langle v_{g},u_{g}\rangle})$$(U_{\langle u_{0},v_{0}\}}, V_{\{v_{0},u_{0}\rangle})$$\langle v(v_{0}, u_{0})|u(u_{0}, v_{0})\}=\{v(q^{t}v_{g}, u_{g})|q^{st}u(q^{s}u_{g}, v_{g})\rangle=q^{st-st}=1$and$\{v(q^{m}v_{0}, u_{0})|u(q^{k}u_{0}, v_{0})\rangle$$=$$\langle q^{-sm}v(q^{m+t}v_{g}, u_{g})|q^{st+kt}u(q^{k+s}u_{g}, v_{9})\rangle$$=$$q^{st+kt-(-sm)-(m+t)(k+s)}$$=$$q^{-n\iota k}$.

We have now shown that the two structures
andare isomorphic. Therefore so are the two structures$(U, V)_{\phi}$ and $(U,V)_{\psi}$ .
$(U_{\langle u_{g},v_{g}\rangle}, V_{\langle v_{g},u_{g}\rangle})$$(U_{\langle u_{0},v_{0}\rangle}, V_{\{v_{0},u_{0}\rangle})$$\blacksquare$

One notice easily that any particular properties of the complex
numbers are not used in the proof of Proposition 4.4. Hence we seethat Proposition 4.4 can be generalized to quantum 2-tori over any
algebraically closed field$F$of characteristic zero.

Corollary 3 Suppose andare isomorphic algebraically closed
fields of characteristic zero. Let $q\in F$ and $q’\in F’$ such that bothand are transcendental andand $\Gamma’=q^{lZ}$ are elementarilyequivalent infinite multiplicative subgroups. Thenandare isomorphic as quantum 2-tori.
$F’$

$F$

$q$

$\Gamma=q^{Z}$$q^{l}$$T_{q}^{2}(F)$$T_{q}^{2},(F’)$

This enable us to describe the isomorphism type of the quantum
2-tori using the language
given below.
$\mathcal{L}_{q}$3.1

The language

From now on we work with an algebraically closed field of characteristic zero andfor some $q\in F$ (say, transcendental).
Let
where$F$$\Gamma=q^{Z}$$\mathcal{L}_{q}=\mathcal{L}_{T_{q}^{2}}=\{U, V, F, \Gamma, U, V, q, u(\cdot, \cdot), v(\cdot, \cdot), sp, 0,1, T_{p}\}$$\bullet$$\bullet$$\bullet$$\bullet$U, V, F,$\Gamma$are unary predicates,

are 4-ary relations,is a constant symbol,
sp is a ternary predicate such that sp
is interpreted as$\gamma\cdot x=y$ for the module operation, (sp stands for the scalar$U,$ $V$$q$$(\gamma, x, y)$multiplication)$\bullet$$\bullet$,binary function symbols
is a ternary relation symbol corresponding to the pairing function.
$u(\cdot,$$T_{p}$$\cdot),$$v(\cdot,$$\cdot)$

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3.2

The first-order theory of$T_{q}^{2}(F)$Recall that we treat the quantum torus
as a 3-sorted structure$(U, V, F)$ . Two operators
andare acting on $F^{*}U$ and V. We
view both $F^{*}U$ andas the following equivalence classes;$(F\cross U)/E$ where for $(x, y),$ $(x’, y’)\in F\cross U$ define$T_{q}^{2}(F)$

$U$

$V$

$F^{*}$$F^{*}V$$F^{*}U\simeq$(11)$(x, y)\sim E(x’, y’)\Leftrightarrow\exists\gamma\in\Gamma(y’=\gamma y\wedge x’=x\gamma^{-1})$

Similarly for$F^{*}$

V.

Two actions on $U$ and V

3.2.1

Two operators
andare acting on $F^{*}U$ and $F^{*}V$ and we treatbothandas 4-ary relations. These actions have the following
$U$

$U$

$V$

$V$

properties;

1.

$\forall u\in U$$]u\in F^{*}(U:u\mapsto uu)$

and$\forall u\in U\exists v\in F^{:}\exists u’\in U(V :u\mapsto vu’\wedge U :u’\mapsto q^{-1}uu’)$

2. $\forall v\in V\exists v\in F^{*}(V:v\mapsto vv)$ and$\forall v\in V\exists u\in F^{;}\exists v’\in V(U:v\mapsto quv’\wedge V:v\mapsto vv)$

We need to translate the above properties into first-order formulas taking into account the equivalence relation (11); first we express
simply that $U$ and V are acting on both $F^{*}U$ and $F^{*}U$ as follows.
$\bullet$$\forall x_{1}\forall u_{1}\forall x_{2}\forall u_{2}\forall x_{1}’\forall u_{1}’\forall x_{2}’\forall u_{2}’\forall x_{1}’(U(x_{1}, u_{1}, x_{2}, u_{2})arrow(x_{1}\in F^{*}\wedge$$u_{1}\in U\wedge x_{2}\in F^{*}\wedge u_{2}\in U))\wedge((U(x_{1}, u_{1}, x_{2}, u_{2})\wedge U(x_{1}’,$$u_{1}’,$ $x_{2}’,$$u_{2}’\wedge$$(x_{1)}u_{1})\sim E(x_{1}’, u_{1}’))arrow(x_{2}, u_{2})\sim E(x_{2}’, u_{2}’))$

This formula corresponds to: $F^{*}Uarrow F^{*}$ U. We need three
more similar formulas expressing : $F^{*}Uarrow F^{*}U,$: $F^{*}Varrow$$F^{*}V$and $V$ : $F^{*}Varrow F^{*}V$ .
$U$

$V$

$U$

Next we write down the above properties 1 and 2;

1. $\forall x\forall u\exists u((x\in F^{*}\wedge u\in U\wedge u\in F^{*})arrow U(x, u, x, uu))$
This formula corresponds to $U:u\mapsto uu$ .
Second part can be translated similarly.
The property 2 is also translated into first-order formulas.80
3.2.2$VU=qUV$We must write down the equation “ $VU=qUV$ ” using 4-ary relations
and . This can be done as follows;
$V$

$U$

$\forall x_{1}\forall u_{1}\forall x_{2}\forall u_{2}\forall x_{3}\forall u_{3}\forall x_{4}\forall u_{4}\forall x_{5}\forall u_{5}(V(x_{1}, u_{1}, x_{2}, u_{2})\wedge V(x_{2}, u_{2}, x_{3}, u_{3})\wedge$$V(x_{1}, u_{1}, x_{4}, u_{4})\wedge U(x_{4}, u_{4}, x_{5}, u_{5})arrow x_{3}=x_{5}\wedge u_{3}=qu_{5})$3.2.3

The pairing

1. $\forall x\forall y\forall z(T_{q}(x, y, z)arrow((x\in V\wedge y\in U\wedge z\in\Gamma)\vee(x\in U\wedge y\in$$V\wedge z\in\Gamma)))$2.

$\forall x\forall y\forall z\forall z’((T_{q}(x, y, z)\wedge T_{q}(y, x, z’))arrow z$. $z’=1)$

then ternary relation does not hold. This
3. If
or
can be translated into$v’\not\in\Gamma\cdot v$$u’\not\in\Gamma\cdot u$$\forall x\in V\forall y\in U\forall z\forall\gamma\in\Gamma\forall x’\in V(x\neq\gamma x’arrow\neg T_{q}(x, y, z))$$\forall x\in V\forall y\in U\forall z\forall\gamma\in\Gamma\forall y’\in U(y\neq\gamma y’arrow\neg T_{q}(x, y, z))$3.2.4

Quantitative properties of pairing functions

Finally we need to take account of quantitative properties of the pairing function (9) and (10) into the theory.

4Remarks

In this note we only give the first-order description of quantum 2description of the tori aretori. Stability issue of the theory anddiscssued in the forthcoming paper.
$\mathcal{L}_{\omega_{1}\omega}$

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John Baldwin, Categoricity, University Lecture Series, vol.50,
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Boris Zilber, Zariski Geometries Geometry from the Logician’s Point of View, Cambridge University Press, 2010

[Z3]

Boris Zilber, A class of quantum Zariski geometries in Model
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