Generalized Frobenius-Schur Indicators and
Kuperberg 3-manifold Invariants
Liang Chang
Texas A&M University
Southern Regional Algebra Conference
March 14, 2015
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Hopf Algebras
Hopf algebra H(µ, ∆, S, 1H , ε) over C is a vector space equipped with
the following compatible maps:
Multiplication m : H ⊗ H → H;
Unit i : C → H.
i(1) = 1H ;
Comultiplication ∆ : H → H ⊗ H.
∆(h) = ∑(h) h(1) ⊗ h(2) = h(1) ⊗ h(2) (Sweedler notation).
((∆ ⊗ id)∆)(h) = ∑(h) h(1) ⊗ h(2) ⊗ h(3) = h(1) ⊗ h(2) ⊗ h(3) ;
Counit ε : H → C;
Antipode S : H → H.
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Cointegral and Integral
Left cointegral ΛL ∈ H: hΛL = ε(h)ΛL ;
Right cointegral ΛR ∈ H: ΛR h = ε(h)ΛR .
Modulus of H: An algebra homomorphism α ∈ H ∗ such that
ΛL h = α(h)ΛL
Left integral λ L ∈ H ∗ : (id ⊗ λ L )∆(h) = λ L (h) · 1H ;
Right integral λ R ∈ H ∗ : (λ R ⊗ id)∆(h) = λ R (h) · 1H .
Comodulus of H: A grouplike element g ∈ H, such that
(id ⊗ λ R )∆(h) = λ R (h)g
For finite dimensional Hopf algebras, their left and right
(co)integrals exist and are unique up to scalar multiplication.
For semisimple Hopf algebras, their cointegrals and integrals are
two-sided.
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Cointegral and Integral
Left cointegral ΛL ∈ H: hΛL = ε(h)ΛL ;
Right cointegral ΛR ∈ H: ΛR h = ε(h)ΛR .
Modulus of H: An algebra homomorphism α ∈ H ∗ such that
ΛL h = α(h)ΛL
Left integral λ L ∈ H ∗ : (id ⊗ λ L )∆(h) = λ L (h) · 1H ;
Right integral λ R ∈ H ∗ : (λ R ⊗ id)∆(h) = λ R (h) · 1H .
Comodulus of H: A grouplike element g ∈ H, such that
(id ⊗ λ R )∆(h) = λ R (h)g
For finite dimensional Hopf algebras, their left and right
(co)integrals exist and are unique up to scalar multiplication.
For semisimple Hopf algebras, their cointegrals and integrals are
two-sided.
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Cointegral and Integral
Left cointegral ΛL ∈ H: hΛL = ε(h)ΛL ;
Right cointegral ΛR ∈ H: ΛR h = ε(h)ΛR .
Modulus of H: An algebra homomorphism α ∈ H ∗ such that
ΛL h = α(h)ΛL
Left integral λ L ∈ H ∗ : (id ⊗ λ L )∆(h) = λ L (h) · 1H ;
Right integral λ R ∈ H ∗ : (λ R ⊗ id)∆(h) = λ R (h) · 1H .
Comodulus of H: A grouplike element g ∈ H, such that
(id ⊗ λ R )∆(h) = λ R (h)g
For finite dimensional Hopf algebras, their left and right
(co)integrals exist and are unique up to scalar multiplication.
For semisimple Hopf algebras, their cointegrals and integrals are
two-sided.
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Cointegral and Integral
Left cointegral ΛL ∈ H: hΛL = ε(h)ΛL ;
Right cointegral ΛR ∈ H: ΛR h = ε(h)ΛR .
Modulus of H: An algebra homomorphism α ∈ H ∗ such that
ΛL h = α(h)ΛL
Left integral λ L ∈ H ∗ : (id ⊗ λ L )∆(h) = λ L (h) · 1H ;
Right integral λ R ∈ H ∗ : (λ R ⊗ id)∆(h) = λ R (h) · 1H .
Comodulus of H: A grouplike element g ∈ H, such that
(id ⊗ λ R )∆(h) = λ R (h)g
For finite dimensional Hopf algebras, their left and right
(co)integrals exist and are unique up to scalar multiplication.
For semisimple Hopf algebras, their cointegrals and integrals are
two-sided.
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Examples of Hopf Algebras: C(G)
For a finite group G, its group algebra C(G) is a Hopf algebra.
For g, h ∈ G,
m(g ⊗ h) = gh,
i(1) = e,
∆(g) = g ⊗ g,
ε(g) = 1,
S(g) = g −1
Integral λ (g) = δg,e
1
Cointegral Λ =
∑ h
|G| h∈G
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Examples of Hopf Algebras: uq sl(2, C)
Let q be an r -th primitive root of unity with r odd.
uq sl(2, C) is generated by E, F and K such that
E r = F r = 0, K r = 1.
The Hopf algebra structure is 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 .
Right integral, two-sided cointegral and comodulus:
r −1
λ (F m E n K j ) = δm,r −1 δn,r −1 δj,1 , Λ = F r −1 E r −1 ∑ K j , g = K 2
j=0
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Gauge Equivalence for Hopf algebras
A new Hopf algebra structure HF on H is obtained by twisting ∆
and S via a 2-cocycle F ∈ H ⊗ H
∆F (h) = F ∆(h)F −1 ,
εF (h) = ε(h),
SF (h) = βF S(h)βF−1
where βF = m((id ⊗ S)(F )).
Two Hopf algebras H and K are said to be gauge equivalent if
∼ K.
there exists a 2-cocycle F such that HF =
Two finite dimensional Hopf algebras H and K are gauge
equivalent if and only if Rep(H) and Rep(K ) are equivalent as
tensor categories.
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Gauge Equivalence for Hopf algebras
A new Hopf algebra structure HF on H is obtained by twisting ∆
and S via a 2-cocycle F ∈ H ⊗ H
∆F (h) = F ∆(h)F −1 ,
εF (h) = ε(h),
SF (h) = βF S(h)βF−1
where βF = m((id ⊗ S)(F )).
Two Hopf algebras H and K are said to be gauge equivalent if
∼ K.
there exists a 2-cocycle F such that HF =
Two finite dimensional Hopf algebras H and K are gauge
equivalent if and only if Rep(H) and Rep(K ) are equivalent as
tensor categories.
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Gauge Invariants for Hopf Algebras
A quantity f (H) is called a gauge invariant if f (H) = f (K ) for all
Hopf algebras K gauge equivalent to H.
Examples:
• dim(H)
• Tr(S)
• Frobenius-Schur indicators
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Gauge Invariants for Hopf Algebras
A quantity f (H) is called a gauge invariant if f (H) = f (K ) for all
Hopf algebras K gauge equivalent to H.
Examples:
• dim(H)
• Tr(S)
• Frobenius-Schur indicators
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Frobenius-Schur Indicators for Hopf algebras
Let H be a finite dimensional semisimple Hopf algebra with integral λ
and cointegral Λ.
[Linchenko-Montgomery, Kashina-Sommerhäuser-Zhu]
The n-th FS-indicator of a H-module V is defined as
νn (V ) = χV (Λ[n] )
where χV is the character of V and Λ[n] = Λ(1) Λ(2) · · · Λ(n) is the
n-th Sweedler power.
In particular, for the regular H-module H,
νn (H) = λ (Λ[n] ) = λ Λ(1) Λ(2) · · · Λ(n)
For finite group algebra C(G),
νn (V ) =
1
∑ χV (g n )
|G| g∈G
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Frobenius-Schur Indicators for Hopf algebras
Let H be a finite dimensional semisimple Hopf algebra with integral λ
and cointegral Λ.
[Linchenko-Montgomery, Kashina-Sommerhäuser-Zhu]
The n-th FS-indicator of a H-module V is defined as
νn (V ) = χV (Λ[n] )
where χV is the character of V and Λ[n] = Λ(1) Λ(2) · · · Λ(n) is the
n-th Sweedler power.
In particular, for the regular H-module H,
νn (H) = λ (Λ[n] ) = λ Λ(1) Λ(2) · · · Λ(n)
For finite group algebra C(G),
νn (V ) =
1
∑ χV (g n )
|G| g∈G
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Generalized Frobenius-Schur Indicators
[Ng-Schauenburg] The (n, r )-th FS-indicator of a H-module V is
defined as
V
νn,r (V ) = Tr (En,r
)
V : Hom (C, V ⊗n ) → Hom (C, V ⊗n ) is certain linear map
where En,r
H
H
defined via graph calculus.
V
f
f
V1
V2
f : V → V1 ⊗ V2
V
f :C→V
V∗
V
V
V∗
∗
ev : V ∗ ⊗ V → C coev : C → V ⊗ V
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Generalized Frobenius-Schur Indicators
V : Hom (C, V ⊗5 ) → Hom (C, V ⊗5 )
E5,1
H
H
7→
f
f
V V V V V
V V V V V
V
ν5 (V ) := ν5,1 (V ) = Tr (E5,1
)
For regular H-module H,
H
ν5 (H) = Tr (E5,1
) = λ Λ(1) Λ(2) Λ(3) Λ(4) Λ(5)
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Generalized Frobenius-Schur Indicators
V : Hom (C, V ⊗5 ) → Hom (C, V ⊗5 )
E5,2
H
H
7→
f
f
V V V V V
V V V V V
V
ν5,2 (V ) := Tr (E5,2
)
For regular H-module H,
H
ν5,2 (H) = Tr (E5,2
) = λ Λ(1) Λ(3) Λ(5) Λ(2) Λ(4)
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Generalized Frobenius-Schur Indicators
V : Hom (C, V ⊗5 ) → Hom (C, V ⊗5 )
E5,2
H
H
7→
f
f
V V V V V
V V V V V
V
ν5,2 (V ) := Tr (E5,2
)
For regular H-module H,
H
ν5,2 (H) = Tr (E5,2
) = λ Λ(1) Λ(3) Λ(5) Λ(2) Λ(4)
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Heegaard Diagram
Any closed 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 a homeomorphism from ∂ H1 to ∂ H2 .
Heegaard Diagram (Yg , {c1L , ..., cgL }, {c1U , ..., cgU }) presents how H1
and H2 are glued along a genus g closed orientable surface Yg .
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Heegaard Diagram
Any closed 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 a homeomorphism from ∂ H1 to ∂ H2 .
Heegaard Diagram (Yg , {c1L , ..., cgL }, {c1U , ..., cgU }) presents how H1
and H2 are glued along a genus g closed orientable surface Yg .
Example: 3-sphere S 3
cU
H
Y1
cL
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Heegaard Diagram
Any closed 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 a homeomorphism from ∂ H1 to ∂ H2 .
Heegaard Diagram (Yg , {c1L , ..., cgL }, {c1U , ..., cgU }) presents how H1
and H2 are glued along a genus g closed orientable surface Yg .
Example: 3-sphere S 3
cL
cU
Y1
cU
cL
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Heegaard Diagram of Lens Space L(5, 1)
cU
cL
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Kuperberg Invariant ZKup (L(5, 1), H)
cU
cL
Λ(1)
Λ(2) Λ(3) Λ(4) Λ(5)
If H is semisimple,
ZKup (L(5, 1), H) = λ Λ(1) Λ(2) Λ(3) Λ(4) Λ(5) = ν5 (H)
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Kuperberg Invariant ZKup (L(5, 2), H)
cU
cL
Λ(1)
Λ(2) Λ(3) Λ(4) Λ(5)
ZKup (L(5, 2), H) = λ Λ(1) Λ(3) Λ(5) Λ(2) Λ(4) = ν5,2 (H)
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Genus Two Kuperberg Invariant
c1U
c1L
c2L
Λ(1)
S(Λ(3) )
S(Λ(2) )
0
Λ0(1) Λ0(2) Λ(3)
c2U
ZKup (M, H) = λ Λ(1) Λ0(2) Λ0(3) λ 0 Λ0(1) S(Λ(2) )S(Λ(3) )
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Kuperberg Invariants and Framing
For general finite dimensional Hopf algebras, Kuperberg invariant
is defined for framed 3-manifold.
A framing on M consists of three orthogonal non-vanishing vector
fields (b1 , b2 , b3 ) on M.
Any framing can be represented on the Heegaard diagram.
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Framed Heegaard Diagram of Lens Space L(5, 1)
b1 :
b2 :
cU
cL
R
R
R
R
ΛR
(1) Λ(2) Λ(3) Λ(4) Λ(5)
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Kuperberg Invariants
In general,
ZKup (M, f , H) =
∑ upper
∏ λ [n]
(Λ[m] )
[m]
· · · S ai T bi (Λ(i) ) · · ·
circles
where T = Adα∗ ◦ S −2 , Λ[m] = ΛR ( α m and λ [n] = g n * ΛR .
The exponents ai , bi , m, n are determined by the rotation of the
Heegaard circles relative to the vector fields b1 and b2 .
For the above framing f = (b1 , b2 ),
R
R
R
R
ZKup (L(5, 1), f , H) = λ R ΛR
(1) Λ(2) Λ(3) Λ(4) Λ(5)
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Kuperberg Invariants and Gauge Invariants
If H is a finite dimensional Hopf algebra and M is the Lens space
L(n, 1). Then there is a framing on M such that
R
R
ZKup (L(n, 1), f , H) = λ R ΛR
(1) Λ(2) · · · Λ(n) = νn (H)
[Kashina-Montgomery-Ng]
Question. Fix a closed 3-manifold M with framing f , is
ZKup (M, f , H) a gauge invariant for any finite dimensional Hopf
algebra H?
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Kuperberg Invariants and Gauge Invariants
[Kerler-Chen, Chang-Wang] If H is a factorizable ribbon Hopf
algebra, M is the Lens space L(n, r ), there is a framing on M such
that ZKup (M, f , H) is a gauge invariant for H.
For instance,
R
2 R
4 R
4 R
2
ZKup (L(5, 2), f , H) = λ R ΛR
(2) Λ(5) S (Λ(3) )S (Λ(1) )S (Λ(4) )g
where g is the comodulus of H.
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Thank You!
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Homology 3-sphere
Λ
(2) Λ(3) Λ(4) (5)
Λ(1) Λ
Λ0(1)Λ0
0 Λ0
0
0
0
0
(2)Λ(3)Λ(4)Λ(5)Λ(6) Λ(7) (8)
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