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Connes and Riemannian Differential Geometryon Finite Groups
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki
Institute of Physics of the Jagiellonian University
Oslo, 5-9 August 2019
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Noncommutative Geometries
[Majid, Beggs] Hopf algebras, quantum groups, differential algebras,...
[Connes] spectral triples, Dirac operators, cyclic cohomology,...
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Differential calculi
Let A be an algebra over a field k. The first-order differential calculus over A is apair (Ω1(A), d) consisting with
A−A−bimodule Ω1(A),linear map d : A −→ Ω1(A)
satisfying the following conditionsΩ1(A) = span{adb : a, b ∈ A},d obeys the Leibniz rule, i.e. d(ab) = adb+ (da)b, for all a, b ∈ A.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Differential calculi
We say that (Ω1(A), d) isconnected if ker d = 0,left-covariant if there exists a left coaction of Hopf algebra A on Ω1(A),δL : Ω
1(A) −→ A⊗ Ω1(A), such that δL(aωb) = ∆(a)δL(ω)∆(b) for alla, b ∈ A and ω ∈ Ω1(A), and δL ◦ d = (id⊗ d) ◦∆,right-covariant if there exists a right coaction of Hopf algebra A on Ω1(A),δR : Ω
1(A) −→ Ω1(A)⊗A, such that δR(aωb) = ∆(a)δR(ω)∆(b) for alla, b ∈ A and ω ∈ Ω1(A), and δR ◦ d = (d⊗ id) ◦∆,bicovariant if is both left- and right- covariant.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Differential calculi
Universal DC: Ω1(A) = kerm, dua = a⊗ 1− 1⊗ a,[Woronowicz, 1989] Classification of bicovariant DCs on Hopf algebras
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Finite groups
G = {g1, ..., gn}, C(G) = 〈eg | g ∈ G, egeh = δg,heg〉
Hopf algebra structure on C(G)
egeh = δg,heg , ∆(eg) =∑h∈G
eh ⊗ eh−1g , ε(eg) = δe,g , S(eg) = eg−1 .
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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DCs on finite groups
[A.Dimakis, F.Müller-Hoissen, 1994]
Any 1-form df on Ω1(C(G)) can be rewritten as df = −[ ∑g 6=h
θg,h, f
], where
θg,h = degeh, g 6= h.
θh,keg = θh,kδh,g , egθh,k = θg,hδh,k
δL(θg,h) =∑p
ep ⊗ θ(p−1g),(p−1h)
θg =∑
h∈Gθhg,h are left invariant in the sense of coaction
θgep = epgθg
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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DCs on finite groups
[A.Dimakis, F.Müller-Hoissen, 1994]
Any 1-form df on Ω1(C(G)) can be rewritten as df = −[ ∑g 6=h
θg,h, f
], where
θg,h = degeh, g 6= h.
θh,keg = θh,kδh,g , egθh,k = θg,hδh,k
δL(θg,h) =∑p
ep ⊗ θ(p−1g),(p−1h)
θg =∑
h∈Gθhg,h are left invariant in the sense of coaction
θgep = epgθg
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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DCs on finite groups
[A.Dimakis, F.Müller-Hoissen, 1994]
Any 1-form df on Ω1(C(G)) can be rewritten as df = −[ ∑g 6=h
θg,h, f
], where
θg,h = degeh, g 6= h.
θh,keg = θh,kδh,g , egθh,k = θg,hδh,k
δL(θg,h) =∑p
ep ⊗ θ(p−1g),(p−1h)
θg =∑
h∈Gθhg,h are left invariant in the sense of coaction
θgep = epgθg
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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DCs on finite groups
[A.Dimakis, F.Müller-Hoissen, 1994]
Any 1-form df on Ω1(C(G)) can be rewritten as df = −[ ∑g 6=h
θg,h, f
], where
θg,h = degeh, g 6= h.
θh,keg = θh,kδh,g , egθh,k = θg,hδh,k
δL(θg,h) =∑p
ep ⊗ θ(p−1g),(p−1h)
θg =∑
h∈Gθhg,h are left invariant in the sense of coaction
θgep = epgθg
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
-
DCs on finite groups
[A.Dimakis, F.Müller-Hoissen, 1994]
Any 1-form df on Ω1(C(G)) can be rewritten as df = −[ ∑g 6=h
θg,h, f
], where
θg,h = degeh, g 6= h.
θh,keg = θh,kδh,g , egθh,k = θg,hδh,k
δL(θg,h) =∑p
ep ⊗ θ(p−1g),(p−1h)
θg =∑
h∈Gθhg,h are left invariant in the sense of coaction
θgep = epgθg
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Actions
(Lgf)(h) := f(hg), Lg(ep) = eg−1p, Lg(θp,q) = θg−1p,g−1q
Lg(θh) = θh
The left-invariance under coaction is the same as left-invariance by mean of groupleft action.Similarly, the right-invariance under coaction is the same asright-invariance by mean of group left action.
δL(ω) =∑p∈G
ep ⊗ Lpω, δR(ω) =∑p∈G
Rpω ⊗ ep
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Actions
(Lgf)(h) := f(hg), Lg(ep) = eg−1p, Lg(θp,q) = θg−1p,g−1q
Lg(θh) = θh
The left-invariance under coaction is the same as left-invariance by mean of groupleft action.
Similarly, the right-invariance under coaction is the same asright-invariance by mean of group left action.
δL(ω) =∑p∈G
ep ⊗ Lpω, δR(ω) =∑p∈G
Rpω ⊗ ep
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Actions
(Lgf)(h) := f(hg), Lg(ep) = eg−1p, Lg(θp,q) = θg−1p,g−1q
Lg(θh) = θh
The left-invariance under coaction is the same as left-invariance by mean of groupleft action.Similarly, the right-invariance under coaction is the same asright-invariance by mean of group left action.
δL(ω) =∑p∈G
ep ⊗ Lpω, δR(ω) =∑p∈G
Rpω ⊗ ep
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Actions
The bicovariant differential calculus is given by a subset H ⊆ G× that isad-invariant, i.e. gHg−1 ⊆ H for all g ∈ G. The forms θg , g ∈ H, are the
left-invariant generating forms and the exterior derivative is df =∑
g∈H[f, θg ].
The connectivity of the calculus is equivalent to the condition that the subset Hgenerates the entire group G, 〈H〉 = G.
If H = G× the calculus is universal.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Actions
The bicovariant differential calculus is given by a subset H ⊆ G× that isad-invariant, i.e. gHg−1 ⊆ H for all g ∈ G. The forms θg , g ∈ H, are the
left-invariant generating forms and the exterior derivative is df =∑
g∈H[f, θg ].
The connectivity of the calculus is equivalent to the condition that the subset Hgenerates the entire group G, 〈H〉 = G.
If H = G× the calculus is universal.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Higher calculi
The extension of the universal first order calculus to higher orders is notunique.
Maximal extension: Ωn(C(G)) :=(Ω1(C(G))
)⊗C(G)nFor the universal calculus:dθg = −
∑p∈G×
(θp ⊗ θg + θg ⊗ θp) +∑
p∈G\{e,g}θp ⊗ θgp−1 .
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Higher calculi
The extension of the universal first order calculus to higher orders is notunique.
Maximal extension: Ωn(C(G)) :=(Ω1(C(G))
)⊗C(G)n
For the universal calculus:dθg = −
∑p∈G×
(θp ⊗ θg + θg ⊗ θp) +∑
p∈G\{e,g}θp ⊗ θgp−1 .
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Higher calculi
The extension of the universal first order calculus to higher orders is notunique.
Maximal extension: Ωn(C(G)) :=(Ω1(C(G))
)⊗C(G)nFor the universal calculus:dθg = −
∑p∈G×
(θp ⊗ θg + θg ⊗ θp) +∑
p∈G\{e,g}θp ⊗ θgp−1 .
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Higher calculi
Take ZN with N > 4 and its generator p, pN = e. Assume that we takeH = {p, p−1}, so the only nonvanishing one-forms are θp and θp−1 .For theminimal nonuniversal calculus we obtain the following rules:
θp ∧ θp = 0, θp−1 ∧ θp−1 = 0
dθp = −(θp ∧ θp−1 + θp−1 ∧ θp) = dθp−1
We are free to impose additional conditions, e.g. dθp = dθp−1 = 0.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Higher calculi
Take ZN with N > 4 and its generator p, pN = e. Assume that we takeH = {p, p−1}, so the only nonvanishing one-forms are θp and θp−1 .For theminimal nonuniversal calculus we obtain the following rules:
θp ∧ θp = 0, θp−1 ∧ θp−1 = 0
dθp = −(θp ∧ θp−1 + θp−1 ∧ θp) = dθp−1
We are free to impose additional conditions, e.g. dθp = dθp−1 = 0.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Metric
We say that g = g(1) ⊗ g(2) ∈ Ω1 ⊗A Ω1 is a Riemannian metric if there exists abimodule map (·, ·) : Ω1 ⊗A Ω1 −→ A such that
(ω,g(1))g(2) = ω = g(1)(g(2), ω)
for all ω ∈ Ω1.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Metric
Let g =∑
g∈Hgabθa ⊗ θb. Then (θa, θb) = δa−1bFa for F ∈ C(G) and a
nondegenerate metric on a left-covariant first order differential calculus overA = C(G) given by H ⊆ G× is of the form
g =∑a∈H
1
Ra−1Fa−1θa ⊗A θa−1 .
Moreover, (g(2),g(1)) = |H|.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Metric
Let g =∑
g∈Hgabθa ⊗ θb. Then (θa, θb) = δa−1bFa for F ∈ C(G) and a
nondegenerate metric on a left-covariant first order differential calculus overA = C(G) given by H ⊆ G× is of the form
g =∑a∈H
1
Ra−1Fa−1θa ⊗A θa−1 .
Moreover, (g(2),g(1)) = |H|.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Linear connection
Definition
A linear map ∇ : Ω1 −→ Ω1 ⊗A Ω1 s.th.:∇(aω) = da⊗A ω + a∇ω∇(ωa) = (∇ω)a+ σ (ω ⊗A da),
where σ : Ω1 ⊗A Ω1 −→ Ω1 ⊗A Ω1.
Lemma
Suppose ∇θg =∑a,b
Γga,bθa ⊗ θb and σ(θg ⊗ θh) =∑a,bψg,ha,b θa ⊗ θb. Then
Γga,b + δgb = ψg,bag−1
a,b
for ba 6= g.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Linear connection
Definition
A linear map ∇ : Ω1 −→ Ω1 ⊗A Ω1 s.th.:∇(aω) = da⊗A ω + a∇ω∇(ωa) = (∇ω)a+ σ (ω ⊗A da),
where σ : Ω1 ⊗A Ω1 −→ Ω1 ⊗A Ω1.
Lemma
Suppose ∇θg =∑a,b
Γga,bθa ⊗ θb and σ(θg ⊗ θh) =∑a,bψg,ha,b θa ⊗ θb. Then
Γga,b + δgb = ψg,bag−1
a,b
for ba 6= g.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Torsion-free connection
Torsion
T∇ = ∧∇− d
Theorem
θg ∧ θh = −∑a∈H
(Γga−1hg,a
+ δg,a)θa−1hg ∧ θa
Example
Consider the case of the universal differential calculus. Then torsion-free conditiontakes the following form dθg =
∑a,b∈G×
(δba,g − δga − δgb)θa ∧ θb.
The universal calculus over C(G) has a unique torsion-free linear connection.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Torsion-free connection
Torsion
T∇ = ∧∇− d
Theorem
θg ∧ θh = −∑a∈H
(Γga−1hg,a
+ δg,a)θa−1hg ∧ θa
Example
Consider the case of the universal differential calculus. Then torsion-free conditiontakes the following form dθg =
∑a,b∈G×
(δba,g − δga − δgb)θa ∧ θb.
The universal calculus over C(G) has a unique torsion-free linear connection.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Torsion-free connection
Torsion
T∇ = ∧∇− d
Theorem
θg ∧ θh = −∑a∈H
(Γga−1hg,a
+ δg,a)θa−1hg ∧ θa
Example
Consider the case of the universal differential calculus. Then torsion-free conditiontakes the following form dθg =
∑a,b∈G×
(δba,g − δga − δgb)θa ∧ θb.
The universal calculus over C(G) has a unique torsion-free linear connection.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
-
Torsion-free connection
Torsion
T∇ = ∧∇− d
Theorem
θg ∧ θh = −∑a∈H
(Γga−1hg,a
+ δg,a)θa−1hg ∧ θa
Example
Consider the case of the universal differential calculus. Then torsion-free conditiontakes the following form dθg =
∑a,b∈G×
(δba,g − δga − δgb)θa ∧ θb.
The universal calculus over C(G) has a unique torsion-free linear connection.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Example
The minimal calculus over ZN , N > 4
Consider the case with dθg = 0. Then the torsion-free connection is determined byΓga,b s.th. Γ
ga,b = Γ
gb,a, where g, a, b ∈ {p, p
−1}.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Metric compatible connections
The connection is called metric-compatible iff
(∇⊗ id)g + (σ ⊗ id)(id⊗∇)g = 0
Suppose g =∑
g∈H Gg θg ⊗ θg−1 . Then the above condition takes the followingform (
(Gc−1 −Ra−1Gc−1 ) δbc−1 +Gc−1Γc−1ab
)+
+∑g 6=ba
Gg(Rg−1Γ
g−1
bag−1,c
)(Γga,b + δgb
)= 0
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Example: Z2 = {e, p}
Let us denote G0 = Gp(e), G1 = Gp(p), Γ0 = Γpp,p(e) and Γ1 = Γ
pp,p(p). Then
G0 = ±G1 and (Γ0 + 1)(Γ1 + 1) = ±1. Moreover, only the constant metric allowsa metric-compatible, torsion-free linear connection.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Relations with Connes’ geometry
Let (A,H,D) be a finite spectral triple. Then
Ω1D(A) :=
{∑k
ak[D, bk] | ak, bk ∈ A}
(1)
together with d : A→ Ω1D(A) given by d(·) = [D, ·] is a first order differentialcalculus with a ∗-structure.
Let us now introduce the matrix representation
ρ(egi ) = Eii, ρ(θjk) = cjkEjk, (2)
with c∗jk = ckj , where Ejk is a matrix with 1 in position (j, k) and zeros elsewhere.Then, the matrix representation for the Dirac operator is of the form
D = −∑i 6=j
ρ(θij). (3)
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Relations with Connes’ geometry
Let (A,H,D) be a finite spectral triple. Then
Ω1D(A) :=
{∑k
ak[D, bk] | ak, bk ∈ A}
(1)
together with d : A→ Ω1D(A) given by d(·) = [D, ·] is a first order differentialcalculus with a ∗-structure.
Let us now introduce the matrix representation
ρ(egi ) = Eii, ρ(θjk) = cjkEjk, (2)
with c∗jk = ckj , where Ejk is a matrix with 1 in position (j, k) and zeros elsewhere.Then, the matrix representation for the Dirac operator is of the form
D = −∑i 6=j
ρ(θij). (3)
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Relations with Connes’ geometry
We have to analyse properties of this operator in the context of left- andright-covariance of the calculus. This is still work in progress.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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Conclusions
There are two main approaches to differential structures on finite geometries- spectral geometry & quantum Riemannian geometry.We analysed structures of qRG in the view-point of possible comparizonwith spectral geometry.It seems that these two approaches may have many common points.It is still work in progress.
Arkadiusz Bochniak, Andrzej Sitarz, Paweł Zalecki Connes and Riemannian DG on Finite Groups
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