Noncommutative Cross-Ratios

0.1 MB·

by  Vladimir Retakh

NONCOMMUTATIVE CROSS-RATIOS
VLADIMIR RETAKH
4
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0 1. Introduction
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b The goal of this note is to present a definition and to discuss basic properties of cross-
e ratios over (noncommutative) division rings or skew-fields. We present the noncommutative
F
cross-ratios as products of quasi-Plu¨cker coordinates introduced in [3] (see also [4]). Actually,

7
noncommutative cross-ratios were already mentioned in a remark in [3]. I decided to return
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to the subject after my colleague Feng Luo explained to me the importance of cross-ratios in

] modern geometry (see, for example, [6, 7, 8]).
A
I have to thank Feng Luo for valuable discussions. This work was partially supported by
R
Simons Collaborative Grant.
.
h
t
a 2. Quasi-Plu¨cker coordinates
m
We recall here only the theory of quasi-Plu¨cker coordinates for 2 × n-matrices over a
[
noncommutative division ring F. For general k×n-matrices the theory is presented in [3, 4].
2
a a
v Recall (see [1, 2] and subsequent papers) that for a matrix 1k 1i one can define four
0 a a
2k 2i
7 (cid:18) (cid:19)
quasideterminants provided the corresponding elements are invertible:
7
5 a a a a
1. a1k a1i = a1k −a1ia−2i1a2k, a1k a1i = a1i −a1ka−2k1a2i,
0 (cid:12) 2k 2i(cid:12) (cid:12) 2k 2i (cid:12)
14 (cid:12)(cid:12)(cid:12) aa1k aa1i(cid:12)(cid:12)(cid:12) = a2k −a2ia−1i1a1k, (cid:12)(cid:12)(cid:12)aa1k aa1i (cid:12)(cid:12)(cid:12) = a2i −a2ka−1k1a1i.
: 2k 2i 2k 2i
v (cid:12) (cid:12) (cid:12) (cid:12)
i a (cid:12) a ...(cid:12) a (cid:12) (cid:12)
X Let A = 11(cid:12) 12 (cid:12) 1n be a matrix ov(cid:12)er F. (cid:12)
a (cid:12) a ...(cid:12) a (cid:12) (cid:12)
r (cid:18) 21 22 2n(cid:19)
a
Lemma 2.1. Let i 6= k. Then
−1 −1 −1
a1k a1i a1k a1j a1k a1i a1k a1j
=
(cid:12)a2k a2i (cid:12) (cid:12)a2k a2j (cid:12) (cid:12)a2k a2i (cid:12) (cid:12)a2k a2j (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
if the corresponding expressions are defined.
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Note that in the formula the boxed elements on the left and on the right must be on the
same level.
Definition 2.2. We call the expressions
−1 −1 −1
qk(A) = a1k a1i a1k a1j = a1k a1i a1k a1j
ij (cid:12)a2k a2i (cid:12) (cid:12)a2k a2j (cid:12) (cid:12)a2k a2i (cid:12) (cid:12)a2k a2j (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
the quasi-Plu¨cker coordi(cid:12)nates of m(cid:12)atr(cid:12)ix A. (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Key words and phrases. cross-ratio,quasideterminant, quasi-Plu¨cker coordinate.
1
2 VLADIMIRRETAKH
Our terminology is justified by the following observation. Recall that in the commutative
case the expressions
a a
pik(A) = a1i a1k = a1ia2k −a1ka2i
2i 2k
(cid:12) (cid:12)
are the Plu¨cker coordinates of A. One(cid:12) can see(cid:12)that in the commutative case
(cid:12) (cid:12)
(cid:12) (cid:12)
p (A)
qk(A) = jk ,
ij p (A)
ik
i.e. quasi-Plu¨cker coordinates are ratios of Plu¨cker coordinates.
Let us list properties of quasi-Plu¨cker coordinates over (noncommutative) division ring F.
We sometimes write qk instead of qk(A) where it cannot lead to a confusion.
ij ij
1). Let g be an invertible matrix over F. Then
qk(g ·A) = qk(A).
ij ij
2) Let Λ = diag (λ1,λ2,...,λn) be an invertible diagonal matrix over F. Then
qk(A·Λ) = λ−1 ·qk(A)·λ .
ij i ij j
3) If j = k then qk = 0; if j = i then qk = 1 (we always assume i 6= k.)
ij ij
4) qk ·qk = qk . In particular, qkqk = 1.
ij jℓ iℓ ij ji
5) “Noncommutative skew-symmetry”: For distinct i,j,k
qk ·qi ·qj = −1.
ij jk ki
One can also rewrite this formula as q kqi = −qj .
ij jk ik
6) “Noncommutative Plu¨cker identity”: For distinct i,j,k,ℓ
qkqℓ +qkqj = 1.
ij ji iℓ ℓi
One can easily check two last formulas in the commutative case. In fact,
p p p
qk ·qi ·qj = jk ki ij = −1
ij jk ki p p pkj
ik ji
because Plu¨cker coordinates are skew-symmetric: p = −p for any i,j.
ij ji
Also, assuming that i < j < k < ℓ
p p p p
qkqℓ +qkqj = jk iℓ + ℓk ij.
ij ji iℓ ℓi p p p p
ik jℓ ik ℓj
Because pℓk = pkℓ, the last expression equals to
p p
ℓj jℓ
p p p p p p +p p
jk iℓ kℓ ij ij kℓ iℓ jk
+ = = 1
p p p pjℓ p p
ik jℓ ik ik jℓ
due to the celebrated Plu¨cker identity
p p −p p +p p = 0.
ij kℓ ik jℓ iℓ jk
Remark 2.3. We presented here a theory of the left quasi-Plu¨cker coordinates for 2 by n
matrices where n > 2. A theory of the right quasi-Plu¨cker coordinates for n by 2 or, more
generally, for n by k matrices where n > k can be found in [3, 4].
NONCOMMUTATIVE CROSS-RATIOS 3
3. Definition and basic properties of cross-ratios
We define cross-ratios over (noncommutative) division ring F by imitating the definition of
calssical cross-ratios in homogeneous coordinates, namely, if the four points are represented
in homogeneous coordinates by vectors a,b,c,d such that c = a+b and d = ka+b, then their
cross-ratio is k.
Let
x y z t
1 1 1 1
x = , y = , z = , t =
x y z t
2 2 2 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
be vectors in F2. We define the cross-ratio κ = κ(x,y,z,t) by equations
t = xα+yβ
z = xαγ +yβγ ·κ
(
where α,β,γ,κ ∈ F.
Consider matrix
x y z t
1 1 1 1
x y z t
2 2 2 2
(cid:18) (cid:19)
and index its columns by x,y,z,t.
Theorem 3.1.
κ(x,y,z,t) = qy ·qx .
zt tz
Proof. We assume that coordinates of vectors x,y,z,t are not equal to zero. Equation t =
xα+yβ leads to the system
t1 = x1α+y1β
.
(t2 = x2α+y2β
The system implies that
−1 −1
y1 x1 y1 t1 x1 y1 x1 t1
(3.1) α = , β =
y2 x2 y2 t2 x2 y2 x2 t2
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(we treat α,β as unknow(cid:12)ns here.(cid:12)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Equation z = xαγ +yβγκ leads to the system
z1 = x1αγ +y1βγκ
.
(z2 = x2αγ +y2βγκ
The system implies that
−1 −1
y x y z x y x z
1 1 1 1 1 1 1 1
(3.2) αγ = , βγκ = .
y x y t x y x z
2 2 2 2 2 2 2 2
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Then formulas (3.1),(cid:12)(3.2) im(cid:12)ply(cid:12) that (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
−1
y1 t1 y1 z1
γ =
y2 t2 y2 z2
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
and (cid:12) (cid:12) (cid:12) (cid:12)
−1(cid:12) (cid:12) (cid:12) (cid:12)−1
κ(x,y,z,t) = y1 z1 y1 t1 · x1 t1 x1 t1 = qy qx .
y2 z2 y2 t2 x2 t2 x2 t2 zt tz
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
4 VLADIMIRRETAKH
(cid:3)
The theorem is proved.
Corollary 3.2. Let x,y,z,t be vectors in F, g be a 2 by 2 matrix over F and λ ∈ F,
i
i = 1,2,3,4. If matrix g and elements λ are invertible then
i
−1
(3.3) κ(gxλ1,gyλ2,gzλ3,gtλ4) = λ3 κ(x,y,z,t)λ3 .
Also, asexpected, inthecommutative casetherighthandsideof(3.3)equalstoκ(x,y,z,t).
A proof immediately follows from the properties of quasi-Plu¨cker coordinates.
Remark 3.3. Note that the group GL2(F) acts on vectors in F2 by multiplication from the
left: (g,x) 7→ gx, and the group F× of invertible elements in F by multiplication from the
right: (λ,x) 7→ xλ−1. It defines the action of GL2(F)×T4(F) on P4 = F2 ×F2 ×F2 ×F2
where T4(F) = (F×)4. The cross-ratios are relative invariants of the action.
The following theorem generalizes the main property of cross-ratios to the noncommutive
case.
Theorem 3.4. Let κ(x,y,z,t) be defined and κ(x,y,z,t) 6= 0,1. Then 4-tuples (x,y,z,t) and
(x′,y′,z′,t′) from P4 belong to the same orbit of GL2(F)×T4(F) if and only if there exists
µ ∈ F× such that
′ ′ ′ ′ −1
(3.4) κ(x,y,z,t) = µ·κ(x,y ,z ,t)·µ .
Proof. If(x,y,z,t)and(x′y,z,t′)belongtothesameorbitthensuchµexistsbecauseκ(x,y,z,t)
is a relative invariant of the action.
Assume now that formula (3.4) is satisfied and that κ = κ(x,y,z,t) 6= 0,1. Note that the
matrix g with columns z,t is invertible. Otherwise, there exists α ∈ F such that t = zα. If
α = 0 then κ = 0. If α 6= 0 then qx = α and qy = α−1. Then κ = 1 which contradicts the
zt tz
assumption on κ. The matrix g′ with columns z′,t′ is also invertible.
By applying g−1 to matrix with columns x,y,z,t and (g′)−1 to the matrix with columns
x′,y′,z,t′ we get matrices
a11 a12 1 0 b11 b12 1 0
A = , B =
a21 a22 0 1 b21 b22 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
where a ,b 6= 0 for i,j = 1,2. Then
ij ij
−1 −1
a12 1 a12 0 a11 0 a11 1
κ(x,y,z,t) = · .
a22 0 a22 1 a21 1 a21 0
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
It implies that
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
−1 −1 −1 −1 −1
κ(x,y,z,t) = 1·(−a12a22)·(−a11a21) ·1 = a12a22a21a11 .
Similarly, κ(x′,y,z,t′) = b12b−221b21b−111 and we have
−1 −1 −1 −1 −1
(3.5) a12a22a21a11 = µ·b12b22b21b11 ·µ .
To finish the proof, it is enough to find invertible elements λ ∈ F, i = 1,2,3,4 such that
i
−1 −1 −1 −1
A = diag(λ3,λ4)·B ·diag(λ1 ,λ2 ,λ3 ,λ4 ) .
It leads to the system
a11λ1 = λ3b11, a12λ2 = λ3b12
(a21λ1 = λ4b21, a22λ2 = λ4b22
NONCOMMUTATIVE CROSS-RATIOS 5
Set λ3 = µ. Then
−1 −1 −1 −1 −1
λ1 = a11µb11, λ2 = a12µb12, λ4 = a21λ1b21 = a21a11µb11
and the system is consistent if
−1 −1 −1 −1 −1
a21a11µb11b21 = a22λ2b22 = a22a12µb12b22
(cid:3)
which leads to formula (3.5) and the proof is finished.
The following corollary shows that the defined cross-ratios satisfy cocycle conditions (see
[5]).
Corollary 3.5. For vectors x,y,z,t,w
κ(x,y,z,t) = κ(w,y,z,t)κ(x,w,z,t)
κ(x,y,z,t) = 1−κ(t,y,z,x)
Proof. By definition
κ(w,y,z,t)·κ(x,w,z,t) = qy qw ·qwqx = qy qx = κ(x,y,z,t) .
zt tz zt tz zt tz
(cid:3)
the second formula follows directly from the noncommutative Plu¨cker identity.
The proposition also implies
Corollary 3.6. For vectors x,x ,x ,...x ,z,t ∈ F2 one has
1 2 n
κ(x,x,z,t) = 1
and
κ(xn−1,xn,z,t)κ(xn−2,xn−1,z,t)...κ(x1,x2,z,t) = κ(x1,xn,z,t)
Proof. Note that
κ(x,x,z,t) = qxqx = 1
zt tz
and
κ(xn−1,xn,z,t)κ(xn−2,xn−1,z,t)...κ(x1,x2,z,t) =
= qzxtnqtxzn−1qzxtn−1qtxzn−2...qzxt2qtxz1 = qzxtnqtxz1 = κ(x1,xn,z,t) .
(cid:3)
.
4. Noncommutative cross-ratios and permutations
Thereare24cross-ratiosdefinedforvectorsx,y,z,t ∈ F2. Theyarerelatedbythefollowing
formulas:
Proposition 4.1. Let x,y,z,t ∈ F. Then
(4.1) qxκ(x,y,z,t)qx = qy κ(x,y,z,t)qy = κ(y,x,t,z);
tz zt tz zt
(4.2) qy κ(x,y,z,t)qy = qt κ(x,y,z,t)qt = κ(z,t,x,y);
xz zx xz zx
(4.3) qx κ(x,y,z,t)qx = qt κ(x,y,z,t)qt = κ(t,z,x,y);
yz zy yz zy
−1
(4.4) κ(x,y,z,t) = κ(y,x,z,t).
6 VLADIMIRRETAKH
Noteagaintheeffectofconjugationinthenoncommutativecasesinceqk andqk areinverses
ij ji
to each other.
Proof. For (4.1) one has
qxκ(x,y,z,t)qx = qx ·qy qx ·qx = qxqy = κ(y,x,t,z).
tz zt tz zt tz zt tz zt
The second part of the formula can be proved in a similar way.
For (4.4)
κ(x,y,z,t)−1 = (qy ·qx)−1 = qx ·qx = κ(y,x,z,t).
zt tz zt tz
For (4.2)
qy κ(x,y,z,t)qy = qy ·qy qx ·qy = qy qxqy =
tz zt xz zt tz zx xt tz zx
= qy (−qz qt )qy = −qy qz ·qt qy =
xt tx xz zx xt tx xz zx
= −(1−qy qt )qt qy = −qt qy +1 = qt qz = κ(z,t,x,y) .
xz zx xz zx xz zx xy yx
Also,
qt κ(x,y,z,t)qt = qt ·qy qx ·qt = qt qy (−qz ) =
xz zx xz zt tz zx xz zt tx
−qt (qy qy )qz = −qt qy (1−qy qt ) =
xz zx xt tx xz zx xz zx
= −qt qy +1 = qt qt = κ(z,t,x,y) .
xz zx xy yx
(cid:3)
One can also prove (4.3) in a similar way.
By using Proposition 4.1 and the cocycle condition one can get all 24 formulas for cross-
ratios of x,y,z,t.
References
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no. 2, pp. 91–102.
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Funct. Anal. Appl. 26 (1992), no. 4, pp. 1–20.
[3] I. Gelfand, V. Retakh, Quasideterminants, I, Selecta Math. 3 (1997), no. 4, pp. 517–546.
[4] I. Gelfand, S. Gelfand, V. Retakh, R. Wilson, Quasideterminants, Advances in Math., 193 (2005), no 1,
pp. 56–141.
[5] F. Labourie, What is a Cross Ratio, Notices of the AMS, 55 (2008), no. 10, pp.1234–1235.
[6] F. Labourie, G. McShane, Cross ratios and identities for higher Teichmller-Thurston theory, Duke Math.
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[7] F. Luo, Volume optimization, normal surface and Thurston equation, J. Differential Geom., 93 (2013),
no. 2, pp. 175–353.
[8] F. Luo, Solving Thurston equation in a commutative ring, arXiv: 1201.2228(2012)
E-mail address: [email protected]
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854, USA