The invariant of Turaev-Viro from Group category

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The invariant of Turaev-Viro from Group category

Jerome Petit

To cite this version:

Jerome Petit. The invariant of Turaev-Viro from Group category. 2006. �hal-00089825v2�

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ccsd-00089825, version 2 - 29 Aug 2006

JÉRÔMEPETIT

Abstrat. AGroupategory isa spherialategory whose simpleobjets

areinvertible. TheinvariantofTuraev-Viro withthispartiular ategory is

infattheinvariantofDijkgraaf-Wittenwhosethegroupandthe3-oyleis

givenbythesimpleobjetsandtheassoiativityonstraintoftheategory.

Contents

Introdution 2

1. TheinvariantofDijkgraaf-Witten 2

2. Groupategory 3

2.1. Denition 4

2.2. SomeresultsonGroupategory 5

2.3. 6j-symbol 6

2.4. 6j-symbolfrom Groupategory 7

3. TheinvariantofTuraev-Viro 9

4. Theequality 10

5. Topologialinterpretationofadmissible oloring 12

5.1. ThefundamentalgroupoïdofT ¹²

5.2. Thegaugeation 13

6. ConstrutionofTQFT 14

6.1. TriangulateTQFT 14

6.2. ConstrutionofTuraev-Viro 14

7. Examples 16

7.1. α= 1 ¹⁶

7.2. G=Z_n ¹⁷

7.3. G=S3 ²⁰

7.4. ExamplesofTQFTs 20

AppendixA. Someomputation ofH³(g, k^⋆) ²²

A.1. Example 22

Aknowledgements 23

Referenes 23

2000MathematisSubjetClassiation. 57N10,18D10,20J06.

Keywordsandphrases. Quantuminvariant,invariantofTuraev-Viro,monoidalategory.

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Introdution

In 1992 M. Wakui [15℄ reformulated the invariant of Dijkgraaf-Witten [4℄ and

he proved the topologial invariane in a rigorous way. The invariane is based

upon the triangulationand the Pahner moves. One given a nite groupand a

3-oyle the Dijkgraaf-Witten invariant is dened ombinatorially. Moreover in

thispaperhebuiltatopologialquantumeldtheory(TQFT)fromthisinvariant.

ThesameyearV.TuraevandO.Viro[12℄ builtaninvariantof3manifoldthanks

to 6-j symbol to prove the topologial invariane they showed a relative version

of atheoremof Alexander[1℄on equivaleneof triangulation. This invariantwas

reformulatedinaategoriallanguages[11℄andtheTQFTwasbuilt. Inthesame

spiritof[11℄J.W.BarretandB.W.Westburry[2℄havebuilta3-manifoldinvariant

usingspherialategories. Inthisonstrutionthetopologialinvarianeputsbak

downthetriangulation andthePahner moves. IndependentlyI.Gelfand andD.

Kazhdan [6℄ have built a 3-manifold using spherial ategories and in 1993 D.N.

YetterhasstudiedanuntwistedversionoftheinvariantofDijkgraaf-Wittenin[16℄

the Turaev-Viro style in . In fat these onstrutions are reformulations of the

Turaev-Viroinvariant. Intherest ofthepaperwewill allsuh kindofinvariant

theinvariantofTuraev-Viroanditwillbedenoted : T VC ^where C ^is^the^ategory

usedtobuild theinvariant.

Themaingoalofthispaperistogivearelationbetweenthistwoapproahesbased

on triangulation. That's why we utilize a "speial" spherial ategory. Roughly

speaking,itis aspherialategorysuhthat everysimpleobjetis invertibleand

hasadimension equalto one. Thedimension isgivenbythe spherialstruture.

In[10℄,F.Quinnalledthisategory: "Groupategory". In[7℄invertibleobjets

arealledsimplesurrentsandthetensorategorywhoseeverysimpleobjetsare

invertibleisdenoted Pointedategory. Theauthorshavedenoted Piardategory

ofC^the^full^tensorâtegoryôfC^whoseôbjetsâre^diret^sumôfînvertibleôbjets

ofC^. ^Thusîf^there îsâ^nite ^numberôf ^simpleôbjetând îfêveryôbjetîs ^nite

diretsumofsimpleobjetthenapointedategoryisequaltoitsPiardategory.

In this paper we will use the terminology of F. Quinn [10℄. L. Crane and D.N.

Yetterhavestudied groupoyleto desribemonoidalategorywithdualsin [3℄.

Hereisanoutlineofthepaper. InSetion1wereallthedenitionoftheDijkgraaf-

Invariant[15℄. In Setion2we givethedenitionand wereall somefats onthe

Group ategory. In Setion 3 we givethe denition of the Turaev-Viroinvariant

of 3-manifold. InSetion 4,we omputethe Turaev-Viroinvariantin the aseof

Group-ategory with other onditionsand we show the main theorem (4.2) . In

setion5wegiveatopologialinterpretationoftheadmissibleolorings. InSetion

6wegivetheonstrutiontheTQFTwhiharisesfrom thisinvariant. Weendthe

paperbydisussingafewexamples.

1. The invariantof Dijkgraaf-Witten

Throughoutthis paperk ^will ^beâômmutativeêld ^suh ^thatcar(k) = 0ând k=k^.

Weuse thedesriptionof[15℄. LetG^be^a^nite ^group,^this ^group^will^be^always

amultipliativegroup. MoreoverkîsârepresentationofG^with^the^trivialâtion.

Thenwean deneZ³(G, k^⋆)^the^setôf ^3-oyleôfG^withôeientsⁱⁿ k^⋆ ând

wexα∈Z³(G, k^⋆)^. ^LetT ^beân^-simplex^withn≥1^,âôlorôfT îs^the^following

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data:

(1.1) γ^: {ôrientedêdgesôf^T} →G,

whihsatisestheonditions:

(i) foranyorientededgee^: γ(e) =γ(e)⁻¹^, ^whereeîs ^theôrientedêdge^with

theoppositeorientation.

(ii) For anyoriented2-simplex(012)ofT ^we^have^: γ(01)γ(12)γ(20) = 1.

WedenoteCol(T)^the^setôfâllôlorsôfT^,îfT îsâtriangulationofan-manifold

M^,^withn≥2^, ^we^denoteCol(M, T)^the^set ôfâllôlorsôfM ^given^byT^. ^When

thereisnoambiguityonthehoieofatriangulation,wedenoteCol(M)^the^set^of

olorsofM.IfM îsâ^manifold^with^boundary^: ∂M^,^then∂M îs êndowed^with â

triangulationwhihomesfromthetriangulationofM^. Îfτ îsâôlorôf∂M ^then

thesetofallolorsofM ^whihêxtendτ^,îs^denotedCol(M, τ)^. ^We^giveânôrder^to

theset ofvertiesofatriangulationofM^, ^thenêah^3-simplex^hasânorientation givenby theasending order. Thenfor γ∈Col(M)ând ^for^the^3-simplex (0123)

weput:

α(∆, γ) =α(γ(01), γ(12), γ(20)),

withα∈Z³(G, k^⋆)^.

Theorem 1.1 (Wakui (92)[15℄). Let G ^be â^nite ^group, ^we ^x â^3-o^yle α∈ Z³(G, k^⋆)^. ^Let M ^be â ômpat ôriented triangulated 3-manifold, T îs â ^trian-

gulation of M^. ^We ^denote ^the ^number ôf ^verties ôf T ^by n0 ând T³ ^the ^set ôf

3-simplex in T^. Âll ^the ^3-simplex âre ôriente^d ^by â ^numbering ôf ^the ^verties.

Given τ∈Col(∂M)^,^we^dene ^the Dijkgraaf-Witten invariantby :

Z_M(τ) =|G|⁻ⁿ⁰ X

γ∈Col(M,τ)

Y

∆∈T³

α(∆, γ)^ǫ^∆,

where

ǫ∆=

1 if∆^and^M ^have^the ^sameorientation,

−1 otherwise.

Then ZM(τ)^does^not^dependôn^the ^hoieôftriangulationofM ând^the^hoieôf

orderof vertiesinM ^whenever^we^x ^atriangulationof∂M ^andτ^.

Thankstotheindependeneofthehoieofnumbering,weanonsideranum-

beringofthetriangulationsuhthatthe3-simplexhavethesameorientationofM^.

Thentheinvariantis:

ZM(τ) =|G|⁻ⁿ⁰ X

γ∈Col(M,τ)

Y

∆∈T³

α(∆, γ),

whereall3-simplex∆^have^the^sameorientationofM^.

Remark 1.2. If we onsider M ^without ^boundary, ^then ZM(∅) ^is ^a ^3-manifold

invariant andwedenote it: ZM^.

2. Group ategory

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2.1. Denition. LetC^beâ^monoidalâtegory^,^byâ^salarôbjet^[14℄ôfC^we^shall

meananobjetofC ^suh^that^: End(X) =k^. ÎfC îsâbelianândkîsalgebraially losed then an objet is salar i it is an simple objet. We denote the set of

isomorphismlassesofsalarobjetsofC^byΛC^.

Denition 2.1. A nitely semisimple monoidal ategory is a monoidal ategory

(C,⊗, I, a, l, r)^suh^that ^:

(a) C îsânâbelian k^-ategory ând⊗îsâ^bifuntork^-linear,

(b) every objetofC îsâ^nite ^diret^sum ôf^salarôbjets ôfC^,

() ♯ΛC <∞ândI îsâ^salarôbjet,

(d) C ^is^sovereign¹

IfCîsâ^nitely^semisimple^monoidalâtegory^,^thenêveryôbjetX ôfC âdmits

arightduality: (X, X^∨, e_X, h_X)ândâ^left^duality^: (X^∨, X, ǫ_X, η_X)^,^weân^take

thesameobjetbeauseC^is ^sovereign. ^By^denition ^of^duality^: eX ^: X⊗X^∨→I

ǫX ^: X^∨⊗X →I ηX ^: I→X⊗X^∨ hX ^: I→X^∨⊗X.

andwehavethefollowingequalities:

(eX⊗idX)(idX⊗hX) =idX

(idX⊗ǫX)(ηX⊗idX) =idX

(idX^∨⊗e_X)(hX⊗id_X^∨) =id_X^∨ (ǫX⊗id_X^∨)(idX^∨⊗η_X) =id_X^∨.

Theleftquantum traeofanendomorphismf ∈EndC(X)^is^dened ^by^: trl(f) =eX(f⊗idX^∨)ηX,

therightquantumtraeofanendomorphismf ∈EndC(X)^is^dened ^by^: trr(f) =ǫX(idX^∨⊗f)hX.

foranyendomorphismsf, gⁱⁿC ^we^have^:

tr_l(f ⊗g) =tr_l(f)trl(g), trr(f ⊗g) =trr(f)trr(g),

trr(f) =trl(f^∨),

themultipliationisgivenbythemultipliationofk=End(I)^.

Denition 2.2. A spherial ategory is a nitely semisimple monoidal ategory

suhthat,for allendomorphism f ⁱⁿC ^we^have ^: trl(f) =trr(f)^.

1Cadmitsarightandaleftdualitywhihareisomorphiasmonoidalfuntor.

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Inaspherialategorywedenotethelefttraebytr^and^so^we^havetr=tr_l=tr_r^.

Thequantum dimensionof anobjetX ⁱⁿâ^spherialâtegoryC îs^dened ^by^: dim(X) =tr(idX),

sowehavedim(X) =dim(X^∨)^.

Denition2.3.

(i) AnobjetX ôfâ^monoidal âtegoryC îsâlledînvertibleⁱ^thereêxistsân

objetY ^suh^thatX⊗Y ∼=I^,^whereI îs^the ^tensorûnit ôfC^.

(ii) Amonoidal ategory isalledpointedievery salar objetisinvertible.

(iii) TheGroup ategoryPi(C⁾ ôf^the^monoidal âtegoryC îs^the^full^monoidal

subategoryof C ^whose ôbjetsâre^diret^sumsôf învertibleôbjets ôf C^.

(iv) AGroupategoryisapointedspherial ategory.

(v) Aθ^-ategoryîsâ^braided,^pointed^nitely ^semisimple^monoidal âtegory.

2.1.1. Example ofGroupategory. Gîsâ^nite^group,^we^denotek[G]^theâtegory

whose objets are G^-graded ^nite dimensional k^-vetor ^spaes² ^and ^whose ^mor-

phismsarek^-linear^morphism^that ^preserves^the^grading. ÎfV ândW âreôbjets

ofk[G]^the^monoidal^struture^ofk[G]^is^given^by^: (V ⊗W)g = X

h, k hk=g

V_h⊗W_k.

theassoiativityistheidentityandtheisomorphismlassesofsalarobjetsarein

bijetionwithG^: g↔δg ^whereδg ^is^denedⁱⁿ ^the^following^way^:

(δg)h=

k ^ifg=h, 0 otherwise.

and every salar objet is invertible, thus k[G] îs â ^Group âtegory. k[G] îs â θ^-ategoryⁱGîsânâbelian^group.

2.2. Some results on Group ategory. Whenever C îs â ^Group âtegory^, ît

followsfrom thedenition ofa Groupategoryand thequantum dimension that

for allX ∈ ΛC ^: dim(X)² = 1^. ^The Grothendiek ring ofC ^is ^isomorphi^to ^the

groupalgebraofthenite groupΛC^,^it^is^denoted K0(C)∼=Z[ΛC]^.

Proposition2.4. If C îsâ^Group âtegory ^then^:

(i) allinvertible objetsare inΛC^,

(ii) (ΛC,⊗, I)^is^a^nite^group.

Proof(i): IfX îsînvertibleⁱⁿC ^then^thereêxists ânôbjetY ⁱⁿC ^suh^that ^: X⊗Y ∼=I^,^thus^we^have^:

X

Z∈ΛC

µ_Z(X)Z⊗Y =I,

whereµZ(X) =dimk(HomC(X, Z))^and^so^we^have^: X

Z^′,Z∈ΛC

µZ(X)µZ^′(Z⊗Y)Z^′=I,

2 G A A

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sineIîsâ^salarôbjet X

Z∈ΛC

µ_Z(X)µZ^′(Z⊗Y) =

1 ^ifZ^′=I, 0 otherwise.

MoreoverHomC(X, Y) îs ^nite^for âllôbjetsⁱⁿ C ând ^soµZ(X)∈N֒→ k^, ^thus

there is only one Z0 ∈ ΛC ^suh ^that µ_Z₀(X) 6= 0 ând ^moreover µ_Z₀(X) = 1 ^so X =Z0∈ΛC^. ^Weân^notie^that îfY îs^theînverseôfX ^thenX⊗Y ∼=Iând ^so

Y ∼=X^∨^.

Proof(ii) : If X îs^salar^then ^by ^denition ôfâ^Group âtegoryX îsînvert-

ible and so there is Y ân ôbjet ôf C ^suh ^that X ⊗Y ∼= I ^we ^have ^seen ^that Y =X^∨∈ΛC^. Înâ^nitely ^salar^monoidalâtegory^we^have^: X^∨⊗X =I⊕Z

whereZ îsânôbjetôfC^,^thus^we^have^: X^∨∼=X^∨⊗I

∼=X^∨⊗(X⊗X^∨)

∼= (X^∨⊗X)⊗X^∨

∼=X^∨⊕Z⊗X^∨

IfX îs ^salar^then X^∨ îs^salar^thusZ⊗X^∨= 0ând X^∨6= 0^. ^Soît^follows^that Z = 0 ând X^∨⊗X ∼=I^, ^then X^∨ îs^the ^left ând^rightînverse ôf X^. ÎfX ând Y

aresalarobjetsthenX⊗Y îsânôbjetôfC ând^:

EndC(X⊗Y)∼=HomC(X, X⊗Y ⊗Y^∨)∼=EndC(X)∼=k,

thenX⊗Y îsâ^salarôbjet^thus(ΛC,⊗, I)îsâ^nite^group.

Theorem 2.5 ([5℄, setion 7.5). Suppose G îsâ ^nite ^group, ^then ^: ^Group âte-

gories withunderlying group G^orrespond^toH³(G, k^⋆)^.

InfatH³(G, k^⋆)^lassiesâll^theassoiativityonstraint(uptomonoidalequiv- alenes). ThegroupG^gives^the^setôf îsomorphi^lassesôf ^salarôbjetsândân

elementα∈H³(G, k^⋆)^gives^theassoiativityonstraintoftheGroupategory. If wetake α, α^′ ∈ Z³(G, k^⋆) ^suh ^that [α] = [α^′] ∈ H³(G, k^⋆) ^then ^we ^obtain ^two

GroupategoriesdenotedbyC(G, α)^andC(G, α^′)^suh^that^: C(G, α)∼=^⊗C(G, α^′)

(monoidalequivalene).

2.3. 6j-symbol. We x D â ^nitely ^monoidal âtegory ^then ^for âll ôbjet X ⁱⁿ D ^we ^have^: X =X1⊕...⊕X_n ^with X_i∈ΛD ^then ^for âll 1 ≥ j ≥ n ^there âre

morphismsi_j ∈HomD(Xj, X)^and p_j ∈HomD(X, Xj)^suh^that p_ji_j =id_X_j ^and P

ji_jp_j=id_X^.

Lemma2.6. Wex a, b, c, d, e, f ∈ΛD ^then^the ^following ^appliation

Ψ^: Hom(a, e⊗d)⊗kHom(e, b⊗c→Hom(a,(b⊗c)⊗d) v⊗w7→(w⊗id_d)v

induesanisomorphism between HomD(a,(b⊗c)⊗d)^and L

e∈ΛHomD(a, e⊗d)⊗kHomD(e, b⊗c)^. ^In ^the ^same^vein^we^have^: HomD(a, b⊗(c⊗d))∼=L

f∈ΛHomD(a, b⊗f)⊗kHomD(f, c⊗d)

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Proof: Bydenition ofD^we^have^: b⊗c=⊕e∈λDµ_e(b⊗c)e^with

µ_e(b⊗c) =dim_k(Hom(e, b⊗c))^. ^Then^for^allf ∈HomD(a,(b⊗c)⊗d)^we^have^: f =idb⊗c⊗iddf

=X

e∈Λ

(iepe⊗idd)f

=X

e∈Λ

(ie⊗id_d)(pe⊗id_d)f,

andsoΨ^is^surjetive. ^Moreover^the^vetor^spaes^are^nitedimensionalandthey havethesamedimensionthuswegettheisomorphism. Theseondisomorphismis

obtainedin thesameway.

a^,^theassoiativityonstraintofD^,^indues^a^naturalisomorphism:

(X⊗Y)⊗Z∼=X⊗(Y ⊗Z)^,^for^allX, Y, Z ∈ob(D)^. ^Then^we^have^the^following

ommutativesquare:

L

e∈ΛHomD(a, e⊗d)⊗kHomD(e, b⊗c) //

∼=

L

f∈ΛHomD(a, b⊗f)⊗kHomD(f, c⊗d)

∼=

HomD(a,(b⊗c)⊗d) _∼

= //HomD(a, b⊗(c⊗d))

thepreviousommutativesquareindues twolinearappliations :

a b c d e f

: HomD(a, e⊗d)⊗kHomD(e, b⊗c)→HomD(a, b⊗f)⊗kHomD(f, c⊗d) a b c

d e f

inv

: HomD(a, b⊗f)⊗kHomD(f, c⊗d)→HomD(a, e⊗d)⊗kHomD(e, b⊗c),

thesearethe6j-symbolofD^.

Wedeneabilinearform inthefollowingway: forallobjetsX, Y^, ω_X,Y ^: HomD(X, Y)⊗HomD(Y, X)→k

f⊗g7→trg(f g).

By denition D ^doesn't âdmit ^negligible ^morphism ^so ω_,^_ îs â non-degenerate bilinearformanditdenesanadjointof

a b c d e f

,forall(a, b, c, de, f)∈ΛD^,

thisadjointisdenotedby:

λ(a, b, c, d, e, f)∈(HomD(e⊗d, a)⊗HomD(b⊗c, e)⊗HomD(a, b⊗f)⊗HomD(f, c⊗d))^⋆.

2.4. 6j-symbolfromGroupategory. IfCîsâ^Groupâtegory^then^forâllX, Y

salarobjetsX⊗Y îsâ^salarôbjet. ^ThusîfX, Y, Z âre^salarôbjets^then^:

(2.1) Hom(Z, X⊗Y)∼=

k ^,^ifX⊗Y ∼=Z 0 , otherwise

IntheaseofGroupategorytheisomorphisms(lemma2.6)beome:

Lemma2.7. Forallsalarobjets (a, b, c, d, e, f)^we ^have^:

HomC(a, e⊗d)⊗kHomC(e, b⊗c)∼=HomC(a,(b⊗c)⊗d)

(2.2)

HomC(a, b⊗f)⊗_kHomC(f, c⊗d)∼=HomC(a, b⊗(c⊗d))

(2.3)

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(i)

a b c d e f

6= 0 ⁱe∼=b⊗c^,a∼= (b⊗c)⊗d^andf ∼=c⊗d

(ii)

a b c d e f

inv

6= 0ⁱ e∼=b⊗c^,a∼=b⊗(c⊗d)^andf ∼=c⊗d

Proof : The assertions (i), (ii) and the isomorphisms (2.2), (2.3) ome from

(2.1). ^Thusⁱⁿ^theâseôf^the^Groupâtegory^the^6j-symbol

a b c d e f

only

depends on b, c, d^. ^Fôr âll ^salarôbjets b, c, d ^we ^put α(b, c, d) =

a b c d e f

.

WeandeneαⁱⁿΛC^. ΛCîsâ^groupând^forg∈ΛC ^we^denoteX_gârepresentation ofthis isomorphismlass,and soforallg, h∈ΛC ^we^have^: X_g⊗X_h ∼=X_gh^. ^By

(2.1)weknowthat HomC(Xgh, X_g⊗X_h) îs âônedimensional vetorspae. For allg, h∈ΛC ^we^putφ(g, h)â^basisôfHomC(Xgh, X_g⊗X_h)^.

Weputg, h, k∈ΛC ând^we^denoteX_g, X_h, X_k ^theirrepresentations. Byonstru- tionα(Xg, X_h, X_k)îsânisomorphismofonedimensionalvetorspaesthusinthe basisφ^we^have^:

(2.4) α(Xg, Xh, Xk)(φ(gh, k)⊗φ(g, h)) =α(g, h, k)(φ(g, hk)⊗φ(h, k)),

with α^: ΛC×ΛC ×ΛC → k^⋆^. ^With ^the^same ^notations^the ^ommutative^square

whihdenesα^indues ^the^following^equality^:

α(g, h, k)(idXg⊗φ(h, k))φ(g, hk) =a(Xg, Xh, Xh)(φ(g, h)⊗idXk)(φ(gh, k)).

Thusα^determine^the^followingisomorphism:

HomC(Xghk,(Xg⊗Xh)⊗Xk)∼=HomC(Xghk, Xg⊗(Xh⊗Xk)) v7→a(Xg, X_h, X_h)v,

a^is^theassoiativityonstraintofC^,^thusa^satises^the^Malane's^pentagon^: ^with Xg, Xh, Xk, Xl^salar^objets

((Xg⊗Xh)⊗Xi)⊗Xj

(Xg⊗(Xh⊗Xi))⊗Xj

Xg⊗((Xh⊗Xi)⊗Xj)

(Xg⊗Xh)⊗(Xi⊗Xj) Xg⊗(Xh⊗(Xi⊗Xj))

a(g,h,i)⊗id

<<

yy yy yy yy yy yy yy

a(g,hi,j)

""

EE EE EE EE EE EE EE

id⊗a(h,i,j)

a(gh,i,j)

a(g,h,ij)//

Ifweapplythelastequalityinthebasisφ^we^have^:

α(g, h, kl)α(gh, k, l) =α(h, k, l)α(g, hk, l)α(g, h, k).

Proposition2.8. If C îsâ ^Group âtegory ^then ^the ^6j-symbol îs^determined ^byâ

3-oyleonZ³(ΛC, k^⋆)ândâ^basis ôf Hom(Xgh, Xg⊗Xh)^.

Therelationof a^with^theîdentityônstraint(r, l)îndues^that ^:

l(h)α(g,1, h) =r(g)^,^for^allg, h∈ΛC ^thusr(g) =α(g,1,1) ^and(g) =α(1,1, h)⁻¹^.

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We anhange α ^suh ^that αîs ^normalized ^without ^hanged ^theohomologous lassofα^. În^termôf^basisφ^, îtîsâ^hangeôf^basis.

3. The invariantof Turaev-Viro

Weadopt theapproahof [6℄ratherthen[12℄,but weuseaspherialategory.

Beauseifweonsiderasovereignategorythereisaproblemin theonstrution.

Theproblemoursatthelevelofindependeneofthenumberingofthe3-simplex.

LetT ^beân^-simplex^withn≥1^,^thenâ^Tûraev-ViroôlorôfT(Turaev-Viropoint ofview)isthefollowingdata: γ ^: {ôrientedêdgesôf^T} →ΛC ^whih^satises^the

onditions:

(i) for anyorientededge e^: γ(e) = γ(e)^∨^, ^where e îs^the ôrientedêdge ^with

theoppositeorientation.

Theset of all Turaev-Viro olorof T ^is ^denoted Col_{T V}(T)^. ^letT ^be^a n^-simplex

and wex anumberingof the verties of F^, êvery^faes ôf T ^hasân orientation givenbytheasendingorder: (012)^. ^Fôrêvery^faes(012)^we^dene^the^following

vetorspae: VC((012), γ) =HomC(I, γ(01)⊗γ(12)⊗γ(20))^.

Lemma3.1.

VC((012), γ)∼=VC((201), γ)∼=VC((120), γ)

(3.1)

VC((012), γ)∼=VC((021), γ)^⋆

(3.2)

Proof(3.1): Itomesfrom thesovereignstrutureofC^.

ForallX, Y, Z∈ob(C)^we^have^:

HomC(I, X⊗Y ⊗Z)↔HomC(I, Y ⊗Z⊗X)

f 7→(ǫX⊗idY⊗Z⊗X)(idX^∨⊗f⊗idX)(hx)

Proof(3.2): ItomesfromthefatthattheategoryC^doesn't^admit^negligible

morphismandsothefollowingbilinearformisnon-degenerate:

contr^: VC((012), γ)⊗VC((021), γ)→k

f⊗g7→f^∨g=tr(f^∨g) =tr(g^∨f)

ThusthevetorspaeVC((012), γ)^isindependentofthestartingpointandifwe hangetheorientationofthe2-simplexweobtainthedualvetorspae. Moreover

thisdualvetorspaeanbeobtainbyahangeofolorinfat:

VC((012), γ)∼=VC((021), γ)^⋆∼=VC((021), γ^′),

withγ^′(02) =γ(01)^,γ^′(21) =γ(12)^,γ^′(10) =γ(20)^. ^LetT ^be^thetriangulationof aompatorientedsurfaeΣândT² ^the^set ôf^2-simplexôfT^,^then^we^dene

VC(Σ, T) = M

γ∈Col(T)

O

f∈T²

VC(f, γ),

andthisspaeisindependentofthehoieofanumberingofT^.

Let∆ ^beâ^3-simplex, â^numberingôf^the^vertiesôf∆ ^givesânorientationof∆^,

withthisorientation∆ ^is^denoted(0123)^. ^We^takeγ∈Col(∆)^and^we^put ^: VC((132), γ)⊗VC((023), γ)⊗VC((031), γ)⊗VC((012), γ)L((0123),γ)

−→ k

(3.3)

v0⊗v1⊗v2⊗v37→dim(γ(13))⁻¹λ(γ(03), γ(01), γ(12), γ(23), γ(02), γ(13)).

3