Onq-deformed Farey sum and a homological interpretation of q-deformed real quadratic irrational numbers

2025-04-27 0 0 440.5KB 34 页 10玖币
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On q-deformed Farey sum and a homological
interpretation of q-deformed real quadratic irrational
numbers
Xin Ren
Abstract
The left and right q-deformed rational numbers were introduced by Bapat, Becker and
Licata via regular continued fractions, and they gave a homological interpretation for left
and right q-deformed rational numbers. In the present paper, we focus on negative continued
fractions and defined left q-deformed negative continued fractions. We give a formula for
computing the q-deformed Farey sum of the left q-deformed rational numbers based on
it. We use this formula to give a combinatorial proof of the relationship between the left
q-deformed rational number and the Jones polynomial of the corresponding rational knot
which was proved by Bapat, Becker and Licata using a homological technique. Finally, we
combine their work and the q-deformed Farey sum, and give a homological interpretation of
the q-deformed Farey sum. We also give an approach to finding a relationship between real
quadratic irrational numbers and homological algebra.
keywords: continued fractions, q-deformed Farey sum, Jones polynomial, real quadratic irra-
tional numbers, 2-Calabi–Yau category
1 Introduction
The notion of q-deformed rational numbers [15] was introduced by Morier-Genoud and Ovsienko
based on some combinatorial properties of rational numbers. They further extended this notion
to arbitrary real numbers [16] by some number-theoretic properties of irrational numbers. These
works are related to many directions including Jones polynomial of rational knots [9, 11, 18, 15],
Teichm¨uller spaces [5], the Markov-Hurwitz approximation theory [8, 10, 13, 22], the modular
group and the Picard group [12, 21], combinators of posets [14, 19, 20] and triangulated cate-
gory [2].
For a formal parameter qand an irreducible fraction r
s, as an enhancement of q-deformed
rational numbers, Bapat, Becker and Licata defined left q-deformed rational number hr
si♭
qand
right q-deformed rational number hr
si♯
qvia the regular continued fractions of r
s, and the right
MSC Classification(2020): 11A55, 05A30, 57K14, 18G80
1
arXiv:2210.06056v5 [math.RT] 3 Jul 2024
q-deformed rational number hr
si♯
qis exactly q-deformed rational number hr
siqconsidered by
Morier-Genoud and Ovsienko. Following [15] and [2], the right q-deformed rational numbers
can be expressed by the right q-deformed regular or negative continued fraction expansions, and
the left q-deformed rational numbers can be expressed by the left q-deformed regular continued
fraction expansions. These q-deformations of the fractions are rational expressions in the variable
qwith integer coefficients. Such as [15, 12], and so on, it may be more convenient from the
perspective of the negative continued fraction expansion when we consider some properties of left
and right q-deformed rational numbers and their applications. In particular, the formula for the
right q-deformed Farey sum based on the negative continued fraction is more concise [15, Section
2]. This induces us to consider the q-deformed Farey sum of the left q-deformed rational numbers.
In the present paper, we define the left q-deformed negative continued fraction expansion. Then
we give a formula for computing the q-deformed Farey sum of the left q-deformed rational
numbers based on negative continued fraction (see Theorem 3.3).
As an application of the right q-deformed rational numbers, given a rational number r
s, we
can use the numerator and denominator of hr
si♯
qto represent the Jones polynomial of the rational
knot to which r
scorresponds [15, Proposition A.1]. On the other hand, Bapat, Becker and Licata
prove that the Jones polynomial for the rational knot corresponding to r
scan be represented by
just the numerator of hr
si♭
q[2, Theorem A.3] by considering a homological interpretation of hr
si♭
q
and hr
si♯
q. Considering the zigzag algebra on the A2quiver, we can obtain a triangulated category
C2called 2-Calabi–Yau category associated to the A2quiver [2, Section 4]. For spherical objects
on C2, Bapat, Becker and Licata defined two functions, denoted as occqand homq, and they
proved that hr
si♭
qand hr
si♯
qcan be expressed in terms of occqand homq, respectively [2, Theorems
3.7 and 3.8]. There are two questions worth considering. Can we give a combinatorial proof of
[2, Theorem A.2] without homology techniques? Can we give a homological interpretation of the
q-deformed irrational numbers defined in [16]? In the present paper, we apply Theorem 3.3 to
give a combinatorial proof of [2, Theorem A.3] without using homology techniques (see Theorem
4.2). Then, we combine the homological interpretation of the left and right q-deformed rational
numbers and the q-deformed Farey sum, and give a homological interpretation of the q-deformed
Farey sum (see Propositions 5.13 and 5.14). We also apply the results in [2, Theorems 4.7 and
4.8] to real quadratic irrational numbers with periodic type (see Theorem 5.15).
This paper is organized into the following sections. In Section 2, we first recall some defini-
tions related to the left and right q-deformed rational numbers, including the (right) q-deformed
Euler continuants, which were introduced by Morier-Genoud and Ovsienko [15]. Similarly, we
define the left q-deformed negative continued fractions and left q-deformed Euler continuants.
We prove that the left q-deformed negative continued fractions and the left q-deformed regular
continued fractions are equal. In Section 3, we give q-deformed Farey sum of left q-rational num-
2
bers based on negative continued fraction, and derive a weighted triangulation and q-deformed
Farey tessellation corresponding to left q-deformed rational numbers. In Section 4, we give a
new proof of [2, Theorem A.2] as an application of q-deformed Farey sum of left q-deformed
rational numbers by induction through the length of the negative continued fraction expansion.
In Section 5, we first combine the results of [2, Theorems 4.7 and 4.8] with the q-deformed
Farey sum to give a homological interpretation of the q-deformed Farey sum. Then we consider
a real quadratic irrational number with periodic type, its q-deformation can also be expressed
in a special form that is related to the q-deformation rational number that approximates it
[12]. Based on the results of these q-deformations, we give the relations between real quadratic
irrational numbers and homological algebra.
2q-deformed continued fractions and q-deformed Euler contin-
uants
In this section, we first briefly review some definitions related to left and right q-deformed
rational numbers (see [2] and [15] for details). We define the left q-deformed negative continued
fraction expansion and introduce the left q-deformed Eulerian continuants. We simply check
that the left q-deformed negative continued fraction expansion is indeed consistent with the left
q-deformed regular continued fraction expansion.
2.1 Left and right q-integers and q-deformed rational numbers
It is well-known that an irreducible fraction r
s∈Q∪ {∞} has unique regular and negative
continued fraction expansions as follows:
r
s=a1+1
a2+1
...+1
a2m
=c1−1
c2−1
...−1
ck
with a1∈Z,ai∈Z\{0}(i≥2), and c1∈Z\{0}(i≥2) and cj∈Z\{−1,0,1}(j≥2). If r
sis
negative, then a1, . . . , a2mand c1, . . . , ckare negative, and if r
sis positive, then a1, . . . , a2mand
c1, . . . , ckare positive. We denote this expansion by [a1, . . . , a2m] and [[c1, . . . , ck]], respectively.
As special cases, the regular and negative continued fraction expansions of 0 and ∞(∞:= 1
0)
are [−1,1], [[1,1]] and empty expansion [ ], [[ ]], respectively.
We consider the following three matrices. σ1:= 1−1
0 1 , σ2:= 1 0
1 1, S := 0−1
1 0 .
We know that the modular group PSL2(Z) can be generated by {σ1, σ2}or {σ1, S}. The modular
3
group PSL2(Z) acts on Q∪ {∞} by the fractional linear transformation:
a b
c d(x) = ax +b
cx +b,
where a b
c d∈PSL2(Z), x∈Q∪ {∞}. Then a rational number r
s= [a1, . . . , a2m] =
[[c1, . . . , ck]] can be expressed by the following formulas:
r
s=σ−a1
1σa2
2σ−a3
1σa4
2···σ−a2m−1
1σa2m
2(∞),(2.1)
r
s=σ−c1
1Sσ−c2
1S···σ−ck
1S(∞).(2.2)
Definition 2.1 ([2]).Let qbe a formal parameter. For a rational number r
s= [a1, . . . , a2m],
we denote by PSL2,q(Z) the subgroup of group GL2(Z[q±1]) generated by the following two
elements:
σ1,q =q−1−q−1
0 1 , σ2,q =1 0
1q−1.
Then the right q-deformed rational number is
hr
si♯
q=σ−a1
1,q σa2
2,qσ−a3
1,q σa4
2,q ···σ−a2m−1
1,q σa2m
1,q (∞),
and the left q-deformed rational number is
hr
si♭
q=σ−a1
1,q σa2
2,qσ−a3
1,q σa4
2,q ···σ−a2m−1
1,q σa2m
1,q 1
1−q.
2.2 Left q-deformed negative continued fractions
Definition 2.2 ([2]).Let qbe a formal parameter. We consider an integer n, the following two
rational forms [n]♭
qand [n]♯
qin qare called the right q-integer of nand the left q-integer of n,
respectively.
[n]♯
q:= 1−qn
1−q,[n]♭
q:= 1−qn−1+qn−qn+1
1−q.
Remark 2.3. Suppose that m, n ∈Z. It can be easy to check that the right q-integers and left
q-integers satisfy the following properties.
(i) [n]♯
q= [n]♭
q+qn−1−qn;
(ii) [m+n]♯
q= [m]♯
q+qm[n]♯
q= [n]♯
q+qn[m]♯
q= [n+n]♯
q,
[m+n]♭
q= [m]♯
q+qm[n]♭
q= [n]♯
q+qn[m]♭
q= [n+m]♭
q;
4
(iii) [−n]♯
q=−q−1[n]♯
q−1, [−n]♭
q=−q−1[n]♭
q−1;
(iv) qn[n]♯
q−1=q[n]♯
q,qn([n]♭
q−1−[0]♭
q−1) = q([n]♭
q−[0]♭
q).
Suppose that r
s= [a1, . . . , a2m] = [[c1, . . . , ck]]. From [15] and [2], the right q-deformed
rational number hr
si♯
qhas both the following q-deformed positive and negative continued fraction
expansions.
hr
si♯
q= [a1, a2, . . . , a2m]♯
q:= [a1]♯
q+qa1
[a2]♯
q−1+q−a2
[a3]♯
q+qa3
[a4]♯
q−1+q−a4
...
[a2m−1]♯
q+qa2m−1
[a2m]♯
q−1
,(2.3)
hr
si♯
q= [[c1, c2, . . . , ck]]♯
q:= [c1]♯
q−qc1−1
[c2]♯
q−qc2−1
[c3]♯
q−qc3−1
[c4]♯
q−qc4−1
...
[ck−1]♯
q−qck−1−1
[ck]♯
q
.(2.4)
For the left q-deformed rational number hr
si♭
q, Bapat,Becker and Licata proved that the right
q-deformed rational number hr
si♭
qhas a q-deformed positive continued fraction expansion [2] as
follows:
5
摘要：

Onq-deformedFareysumandahomologicalinterpretationofq-deformedrealquadraticirrationalnumbersXinRenAbstractTheleftandrightq-deformedrationalnumberswereintroducedbyBapat,BeckerandLicataviaregularcontinuedfractions,andtheygaveahomologicalinterpretationforleftandrightq-deformedrationalnumbers.Inthepresen...

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