Factorizable ribbon Hopf algebras and their
applications in quantum topology
Liang Chang
University of California, Santa Barbara
Texas A&M University
December 4, 2012
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Motivation
Ribbon tensor category
Modular category
Link invariant
Reshetikhin-Turaev invariant
Unimodular ribbon Hopf algebra
Link invariant
Factorizable ribbon Hopf algebra
Hennings invariant
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Motivation
Ribbon tensor category
Modular category
Link invariant
Reshetikhin-Turaev invariant
Unimodular ribbon Hopf algebra
Link invariant
Factorizable ribbon Hopf algebra
Hennings invariant
2 / 33
Hopf Algebras
Hopf algebra H(µ, ∆, S, 1, ε) over C is a complex vector space
equipped with the following compatible maps:
Multiplication µ : H ⊗ H → H;
Unit i : C → H.
i(1) = 1H ;
Comultiplication ∆ : H → H ⊗ H.
P
∆(x) = x(1) ⊗ x(2) (Sweedler notation);
(x)
Counit ε : H → C;
Antipode S : H → H.
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Integral
Left integral ΛL for H: hΛL = ε(h)ΛL ;
Right integral ΛR for H: ΛR h = ε(h)ΛR .
Left integral λ L for H ∗ : φ λ L = φ (1)λ L ;
Right integral λ R for H ∗ : ΛR φ = φ (1)ΛR .
Distinguish grouplike element g of H: φ λ R = φ (g)λ R .
Distinguish grouplike element α of H ∗ : ΛL h = α(h)ΛL .
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Integral
Left integral ΛL for H: hΛL = ε(h)ΛL ;
Right integral ΛR for H: ΛR h = ε(h)ΛR .
Left integral λ L for H ∗ : φ λ L = φ (1)λ L ;
Right integral λ R for H ∗ : ΛR φ = φ (1)ΛR .
Distinguish grouplike element g of H: φ λ R = φ (g)λ R .
Distinguish grouplike element α of H ∗ : ΛL h = α(h)ΛL .
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Facts about integrals
Theorem (Sweedler)
For finite dimensional Hopf algebras, the ideals of left/right
integrals are one-dimensional.
Definition
Unimodular Hopf algebra: the space of left integrals for H is the
same as the space of right integrals for H.
For a unimodular Hopf algebra, the distinguish grouplike
element α = ε.
The right integral λ satisfies λ (xy ) = λ (S 2 (y )x).
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Facts about integrals
Theorem (Sweedler)
For finite dimensional Hopf algebras, the ideals of left/right
integrals are one-dimensional.
Definition
Unimodular Hopf algebra: the space of left integrals for H is the
same as the space of right integrals for H.
For a unimodular Hopf algebra, the distinguish grouplike
element α = ε.
The right integral λ satisfies λ (xy ) = λ (S 2 (y )x).
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Quasitriangular Hopf Algebras
Definition
Quasitriangular Hopf algebra H: there exists an R-matrix
P
R = i si ⊗ ti ∈ H ⊗ H such that
∆op (h) = R∆(h)R −1
(∆ ⊗ id)(R) = R13 R23
(id ⊗ ∆)(R) = R13 R12
Rep(H) is a braided tensor category.
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Quasitriangular Hopf Algebras
Definition
Quasitriangular Hopf algebra H: there exists an R-matrix
P
R = i si ⊗ ti ∈ H ⊗ H such that
∆op (h) = R∆(h)R −1
(∆ ⊗ id)(R) = R13 R23
(id ⊗ ∆)(R) = R13 R12
Rep(H) is a braided tensor category.
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Factorizable Hopf Algebras
Definition
A quasitriangular Hopf algebra H is said to be factorizable if the
Drinfeld map D : H ∗ → H given by
X
D(η) = (η ⊗ id)(R21 R) =
η(ti sj0 )si tj0
i,j
is a linear isomorphism, where R =
P
i si
⊗ ti =
0
0
j sj ⊗ tj .
P
Let Λ, λ be the nonzero right integrals for H and H ∗ ,
respectively. Then H is factorizable if and only if D(λ ) = cΛ
for some non-zero scalar c ∈ C.
Factorizable Hopf algebras are unimodular.
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Factorizable Hopf Algebras
Definition
A quasitriangular Hopf algebra H is said to be factorizable if the
Drinfeld map D : H ∗ → H given by
X
D(η) = (η ⊗ id)(R21 R) =
η(ti sj0 )si tj0
i,j
is a linear isomorphism, where R =
P
i si
⊗ ti =
0
0
j sj ⊗ tj .
P
Let Λ, λ be the nonzero right integrals for H and H ∗ ,
respectively. Then H is factorizable if and only if D(λ ) = cΛ
for some non-zero scalar c ∈ C.
Factorizable Hopf algebras are unimodular.
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Examples of Factorizable Hopf Algebras
C[Zn ] with R-matrix R =
1
n
n
P
e−
2πiab
n
ga ⊗ gb.
a,b=0
C[Zn ] is factorizable iff n is odd.
The Drinfeld double D(H) of a finite dimensional Hopf
algebra H is factorizable.
uq sl(2, C) at an odd root of unity.
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Examples of Factorizable Hopf Algebras
C[Zn ] with R-matrix R =
1
n
n
P
e−
2πiab
n
ga ⊗ gb.
a,b=0
C[Zn ] is factorizable iff n is odd.
The Drinfeld double D(H) of a finite dimensional Hopf
algebra H is factorizable.
uq sl(2, C) at an odd root of unity.
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Examples of Factorizable Hopf Algebras
C[Zn ] with R-matrix R =
1
n
n
P
e−
2πiab
n
ga ⊗ gb.
a,b=0
C[Zn ] is factorizable iff n is odd.
The Drinfeld double D(H) of a finite dimensional Hopf
algebra H is factorizable.
uq sl(2, C) at an odd root of unity.
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Factorizable Ribbon Hopf Algebras
uq sl(2, C) at an odd root of unity. Let q be an r -th primitive
root of unity with r odd. It is generated by E, F and K with
the following relations:
E r = F r = 0, K r = 1
and the Hopf algebra structure given by
KE = q 2 EK , KF = q −2 FK , [E, F ] =
K − K −1
,
q − q −1
∆(E) = 1 ⊗ E + E ⊗ K , ∆(F ) = K −1 ⊗ F + F ⊗ 1, ∆(K ) = K ⊗ K ,
ε(E) = ε(F ) = 0, ε(K ) = 1,
S(E) = −EK −1 , S(F ) = −KF , S(K ) = K −1 .
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Factorizable Ribbon Hopf Algebras
The right integral for uq sl(2, C)∗ , two-sided integral and
distinguish grouplike element are
m
n
j
λ (F E K ) = δm,r −1 δn,r −1 δj,1 , Λ = F
r −1
E
r −1
r −1
X
Kj, g = K2
j =0
The R-matrix is
R=
1
r
X
0≤m,i,j≤r −1
(q − q −1 )m m(m−1)/2+2m(i−j )−2ij m i
q
E K ⊗ F mK j
[m]!
The ribbon element is
θ=
r −1
1 X s2
(
q )(
r
s=0
X
0≤m,j≤r −1
(q −1 − q)m − 1 m+mj + 1 (j +1)2 m m j
2
q 2
F E K )
[m]!
and balancing element is G = K .
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Ribbon Hopf Algebras
Definition
Ribbon Hopf algebra: Quasitrangular Hopf algebra together with
a ribbon element θ satisfying the relations
∆(θ ) = (R21 R)−1 (θ ⊗ θ ), ε(θ ) = 1, and S(θ ) = θ .
Rep(H) is a ribbon tensor category.
P
In a ribbon Hopf algebra, G = ( i S(ti )si )θ −1 called
balancing element is a grouplike element and
S 2 (h) = GhG−1 for h ∈ H.
G2 = g, distinguish grouplike element of H.
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Ribbon Hopf Algebras
Definition
Ribbon Hopf algebra: Quasitrangular Hopf algebra together with
a ribbon element θ satisfying the relations
∆(θ ) = (R21 R)−1 (θ ⊗ θ ), ε(θ ) = 1, and S(θ ) = θ .
Rep(H) is a ribbon tensor category.
P
In a ribbon Hopf algebra, G = ( i S(ti )si )θ −1 called
balancing element is a grouplike element and
S 2 (h) = GhG−1 for h ∈ H.
G2 = g, distinguish grouplike element of H.
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SL(2, Z) action on the center
There is an action of SL(2, Z) on a factorizable ribbon Hopf
algebra H. Let S, T : H → H be given by, for all x ∈ H,
−1
), T x = θ x.
S(x) = (id ⊗ λ )(R −1 (1 ⊗ x)R21
Then
(ST )3 = cS 2 ,
S 2 = S −1 .
for some normalization c ∈ C.
When restricted on the center Z, S 4 |Z = S −2 |Z = id|Z . So
we obtain a projective representation of SL(2, Z) on the
center Z.
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Quantum topology
Dehn surgery
Hennings invariant
Heegaard splitting
Kuperberg invariant
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Dehn Surgery
Theorem (Lickorish)
Every closed, orientable, connected 3-manifold is obtained by
performing Dehn surgery on a framed link in the 3-sphere.
Given a framed link L in S 3 , remove an open tubular
neighborhood T of L.
Glue in solid tori by homeomorphism of their boundaries to
the torus boundary component of S 3 \ T .
Two 3-manifolds S 3 (L) and S 3 (J) are homeomorphic if and
only if L and J are related by a sequence of Kirby moves.
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Dehn Surgery
Theorem (Lickorish)
Every closed, orientable, connected 3-manifold is obtained by
performing Dehn surgery on a framed link in the 3-sphere.
Given a framed link L in S 3 , remove an open tubular
neighborhood T of L.
Glue in solid tori by homeomorphism of their boundaries to
the torus boundary component of S 3 \ T .
Two 3-manifolds S 3 (L) and S 3 (J) are homeomorphic if and
only if L and J are related by a sequence of Kirby moves.
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Dehn Surgery
Theorem (Lickorish)
Every closed, orientable, connected 3-manifold is obtained by
performing Dehn surgery on a framed link in the 3-sphere.
Given a framed link L in S 3 , remove an open tubular
neighborhood T of L.
Glue in solid tori by homeomorphism of their boundaries to
the torus boundary component of S 3 \ T .
Two 3-manifolds S 3 (L) and S 3 (J) are homeomorphic if and
only if L and J are related by a sequence of Kirby moves.
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Hennings Invariant
Hennings invariant of 3-manifolds for unimodular ribbon
Hopf algebra H with λ (θ )λ (θ −1 ) 6= 0
ZHenn (M(L), H) = [λ (θ )λ (θ −1 )]−
c(L)
2
λ (θ ) − σ (L)
2 TR(L, H)
λ (θ −1 )
where c(L) denotes the number of components of L, and
σ (L) denotes the signature of the framing matrix of L.
If H is factorizable, λ (θ −1 ) = λ (θ )−1 6= 0.
ZHenn (M(L), H) = λ (θ )−σ (L) TR(L, H)
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Hennings Invariant
Hennings invariant of 3-manifolds for unimodular ribbon
Hopf algebra H with λ (θ )λ (θ −1 ) 6= 0
ZHenn (M(L), H) = [λ (θ )λ (θ −1 )]−
c(L)
2
λ (θ ) − σ (L)
2 TR(L, H)
λ (θ −1 )
where c(L) denotes the number of components of L, and
σ (L) denotes the signature of the framing matrix of L.
If H is factorizable, λ (θ −1 ) = λ (θ )−1 6= 0.
ZHenn (M(L), H) = λ (θ )−σ (L) TR(L, H)
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Hennings Link Invariant
↔
si
ti
↔
si
;
x = S(x)
;
y
x
=
=
x
S −1 (ti )
S(x)
xy
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Hennings Link Invariant
Slide all the Hopf algebra elements on the same component
into one vertical portion of the diagram. Along a vertical
line, all the Hopf algebra elements are multiplied together to
get a product wi ∈ H for the i-th component.
For a unimodular ribbon Hopf algebra H,
TR(L, H) = λ (w1 Gd1 +1 ) · · · λ (wn Gdn +1 )
is a regular isotopy invariant of links, where di is the
Whitney degree of the i-th component.
wi
d =1
wi
wi
d = −1
d =0
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Hennings Link Invariant
Slide all the Hopf algebra elements on the same component
into one vertical portion of the diagram. Along a vertical
line, all the Hopf algebra elements are multiplied together to
get a product wi ∈ H for the i-th component.
For a unimodular ribbon Hopf algebra H,
TR(L, H) = λ (w1 Gd1 +1 ) · · · λ (wn Gdn +1 )
is a regular isotopy invariant of links, where di is the
Whitney degree of the i-th component.
wi
d =1
wi
wi
d = −1
d =0
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Example of Hennings Link Invariant
Hopf link:
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Example of Hennings Link Invariant
s1
t1
s2
t2
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Example of Hennings Link Invariant
t 2 s1
s2 t 1
TR(L, H) = λ (t 2 s1 )λ (s2 t 1 G2 ) = λ (D(λ )G2 ).
TR(L, H) = λ (Λ) 6= 0 if H is factorizable.
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Example of Hennings Link Invariant
t 2 s1
s2 t 1
TR(L, H) = λ (t 2 s1 )λ (s2 t 1 G2 ) = λ (D(λ )G2 ).
TR(L, H) = λ (Λ) 6= 0 if H is factorizable.
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Example of Hennings Link Invariant
t 2 s1
s2 t 1
TR(L, H) = λ (t 2 s1 )λ (s2 t 1 G2 ) = λ (D(λ )G2 ).
TR(L, H) = λ (Λ) 6= 0 if H is factorizable.
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Example of Hennings Link Invariant
Surgery link of RP 3 #RP 3 :
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Example of Hennings Link Invariant
s5
s1
t5
t1
s6
s4
S −1 (t 4 )
s3
s2
t6
t2
S −1 (t 3 )
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Example of Hennings Link Invariant
s5
s1
t5
t1
S −1 (t 4 )
s2
S −1 (t 3 )
s6
t6
t2
s4
s3
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Example of Hennings Link Invariant
s4 t 1 t 5 S −1 (s5 )S −1 (s6 )s3 t 2 t 6
t 4 t 3 s2 s1
TR(L, H) = λ (t 4 t 3 s2 s1 )λ (s4 t 1 t 5 S −1 (s5 )S −1 (s6 )s3 t 2 t 6 ).
TR(L, H) = λ (Λ(2) Λ(1) ) = tr (S −1 ) if H is factorizable.
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Example of Hennings Link Invariant
s4 t 1 t 5 S −1 (s5 )S −1 (s6 )s3 t 2 t 6
t 4 t 3 s2 s1
TR(L, H) = λ (t 4 t 3 s2 s1 )λ (s4 t 1 t 5 S −1 (s5 )S −1 (s6 )s3 t 2 t 6 ).
TR(L, H) = λ (Λ(2) Λ(1) ) = tr (S −1 ) if H is factorizable.
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Example of Hennings Link Invariant
s4 t 1 t 5 S −1 (s5 )S −1 (s6 )s3 t 2 t 6
t 4 t 3 s2 s1
TR(L, H) = λ (t 4 t 3 s2 s1 )λ (s4 t 1 t 5 S −1 (s5 )S −1 (s6 )s3 t 2 t 6 ).
TR(L, H) = λ (Λ(2) Λ(1) ) = tr (S −1 ) if H is factorizable.
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Heegaard Splitting
Any compact orientable 3-manifold can be decomposed
into the form: H1 ∪f H2 , where H1 and H2 are handlebodies
of the same genus and f is an homeomorphism from the
boundary of H1 to the boundary of H2 .
The common boundary Yg of H1 and H2 is called the
Heegaard surface.
Heegaard diagram is a triple (Yg , {c1l , ..., cgl }, {c1u , ..., cgu }).
{c1l , ..., cgl } are the attaching circles for H1 and called lower
circles.
{c1u , ..., cgu } are the attaching circles for H2 and called upper
circles.
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Heegaard Splitting
Any compact orientable 3-manifold can be decomposed
into the form: H1 ∪f H2 , where H1 and H2 are handlebodies
of the same genus and f is an homeomorphism from the
boundary of H1 to the boundary of H2 .
The common boundary Yg of H1 and H2 is called the
Heegaard surface.
Heegaard diagram is a triple (Yg , {c1l , ..., cgl }, {c1u , ..., cgu }).
{c1l , ..., cgl } are the attaching circles for H1 and called lower
circles.
{c1u , ..., cgu } are the attaching circles for H2 and called upper
circles.
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Heegaard Splitting
Any compact orientable 3-manifold can be decomposed
into the form: H1 ∪f H2 , where H1 and H2 are handlebodies
of the same genus and f is an homeomorphism from the
boundary of H1 to the boundary of H2 .
The common boundary Yg of H1 and H2 is called the
Heegaard surface.
Heegaard diagram is a triple (Yg , {c1l , ..., cgl }, {c1u , ..., cgu }).
{c1l , ..., cgl } are the attaching circles for H1 and called lower
circles.
{c1u , ..., cgu } are the attaching circles for H2 and called upper
circles.
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Heegaard Splitting
Any compact orientable 3-manifold can be decomposed
into the form: H1 ∪f H2 , where H1 and H2 are handlebodies
of the same genus and f is an homeomorphism from the
boundary of H1 to the boundary of H2 .
The common boundary Yg of H1 and H2 is called the
Heegaard surface.
Heegaard diagram is a triple (Yg , {c1l , ..., cgl }, {c1u , ..., cgu }).
{c1l , ..., cgl } are the attaching circles for H1 and called lower
circles.
{c1u , ..., cgu } are the attaching circles for H2 and called upper
circles.
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Heegaard Splitting
Any compact orientable 3-manifold can be decomposed
into the form: H1 ∪f H2 , where H1 and H2 are handlebodies
of the same genus and f is an homeomorphism from the
boundary of H1 to the boundary of H2 .
The common boundary Yg of H1 and H2 is called the
Heegaard surface.
Heegaard diagram is a triple (Yg , {c1l , ..., cgl }, {c1u , ..., cgu }).
{c1l , ..., cgl } are the attaching circles for H1 and called lower
circles.
{c1u , ..., cgu } are the attaching circles for H2 and called upper
circles.
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Examples of Heegaard Diagrams
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Kuperberg Invariant
For a finite dimensional Hopf algebra H and a closed
3-manifold with framing f , Kuperberg constructed an
invariant ZKup (M, f , H) using the Heegaard diagram of M.
A framing of M consists of three linearly independent vector
fields {b1 , b2 , b3 } whose orientation agrees with that of the
manifold.
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Kuperberg Invariant
upper circles
Λ0(2)
λ
Λ(1)
lower circles
Λ
Λ0(3)
Λ(2)
λ
Λ0(1)
Λ0
Λ0(4)
ZKup =
λ (S a1 (Λ(1) )S a2 (Λ(2) )S a3 (Λ0(1) )S a4 (Λ0(2) )g n )λ (S b1 (Λ0(3) )S b2 (Λ0(4) )g m )
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Kuperberg Invariant
upper circles
Λ0(2)
λ
Λ(1)
lower circles
Λ
Λ0(3)
Λ(2)
λ
Λ0(1)
Λ0
Λ0(4)
ZKup =
λ (S a1 (Λ(1) )S a2 (Λ(2) )S a3 (Λ0(1) )S a4 (Λ0(2) )g n )λ (S b1 (Λ0(3) )S b2 (Λ0(4) )g m )
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For each point p on some circle with base point o, ψ(p) is
defined to be the counterclockwise rotation of the tangent to
c relative to b1 from o to p in units of 1 = 360◦ .
φ (p) is defined to be the counterclockwise rotation of the
tangent to c relative to b2 from o to p in units of 1 = 360◦ .
λm− 1 (x) = λ (xg m ), where m = ψ(ol ) the total rotation of the
2
lower circle.
Λn− 1 = (id ⊗ α n )∆(Λ), where −n = ψ(ou ) the total rotation
2
of the upper circle.
ai = 2(ψl (i) − ψu (i)) − 12 .
bi = φl (p) − φu (p)
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Theorem
Let H be a finite dimensional factorizableribbon Hopf algebra
and L(p, q) be an oriented lens space. Then
ZKup (L(p, q), f , H) = ZHenn (L(p, q)#L(p, q), H).
Corollary
|ZKup | = |ZHenn |2
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Lens Space L(p, q)
The three-dimensional lens spaces L(p, q) are quotients of S 3
by Z/p-actions. More precisely, let p and q be coprime integers
and consider S 3 as the unit sphere in C2 . Then the Z/p-action
on S 3 generated by
[e2πi/p ].(z1 , z2 ) := (e2πi/p · z1 , e2πiq/p · z2 )
is free as p and q were coprime. The resulting quotient space is
called the lens space L(p, q).
L(1, 1) = S 3 .
L(2, 1) = RP 3 .
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Future directions
ZKup (M, f , H) = ZHenn (M#M, H) for general 3-manifold M.
Conjecture (Keller)
Given a closed oriented 3-manifold M and a finite dimensional
Hopf algebra H with Drinfeld double D(H) ribbon, there is a
framing f of M such that ZKu (M, f , H) = ZHenn (M, D(H)).
Lattice model realization for Hennings and Kuperberg
invariants.
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Future directions
ZKup (M, f , H) = ZHenn (M#M, H) for general 3-manifold M.
Conjecture (Keller)
Given a closed oriented 3-manifold M and a finite dimensional
Hopf algebra H with Drinfeld double D(H) ribbon, there is a
framing f of M such that ZKu (M, f , H) = ZHenn (M, D(H)).
Lattice model realization for Hennings and Kuperberg
invariants.
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Future directions
ZKup (M, f , H) = ZHenn (M#M, H) for general 3-manifold M.
Conjecture (Keller)
Given a closed oriented 3-manifold M and a finite dimensional
Hopf algebra H with Drinfeld double D(H) ribbon, there is a
framing f of M such that ZKu (M, f , H) = ZHenn (M, D(H)).
Lattice model realization for Hennings and Kuperberg
invariants.
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Thank You!
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