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Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Lecture 3
Daniel Bump
May 24, 2019
W
V ∗ W V
W
V ∗ W V
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Monoidal categories
The axioms for a braided monoidal category are due to Joyaland Street. Braided Monoidal Category (Wikipedia Link)
Let C be a monoidal category. We recall that if A,B,C areobjects in C then there are natural isomorphisms(A⊗ B)⊗ C ∼= A⊗ (B ⊗ C). We will not distinguish betweenthese objects and just denote either as A⊗ B ⊗ C. It is aconsequence of Mac Lane’s coherence theorem that thisidentification will never lead to any difficulties. We will neverworry about this point.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Braided categories
In a braided category there are explicit braid isomorphismscA,B : A⊗ B → B ⊗ A but now we must be careful. For examplethe composition cB,AcA,B is not assumed to be the identity. SocA,B and c−1
B,A are distinct isomorphisms A→ B.
We will notate the morphism cA,B by an over crossing and cB,Aby an under crossing.
A B
B A
cA,B
A B
B A
c−1B,A
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Naturality
We review the important notion of a natural transformation. Weused this implicitly when we defined a monoidal category inLecture 1, where we said that the isomorphisms
(A⊗ B)⊗ C ∼= A⊗ (B ⊗ C)
are required to be natural.
This means, explicitly, the following. Since ⊗ is a bifunctor, ifα : A→ A′, β : B → B′ and γ : C → C′ are morphisms then wehave on the left and right of the following diagram.
(A⊗ B)⊗ C A⊗ (B ⊗ C)
(A′ ⊗ B′)⊗ C′ A′ ⊗ (B′ ⊗ C′)
∼=
(α⊗β)⊗γ α⊗(β⊗γ)
∼=
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Naturality of the commutativity map
The first axiom of a braided category is that the morphismscA,B : A⊗ B → B ⊗ A are to be natural. This means that ifα : A→ A′ and β : B → B′ are morphisms, then
(β ⊗ α) ◦ cA,B = cA′,B′ ◦ (α⊗ β)
α β
A B
B′ A′
= β α
A B
B′ A′
(We are representing the morphisms α, β by dots.)
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Braided category axioms
The braid morphism cA,B : A⊗ B → B ⊗ A is sometimes calledan R-matrix. It is assumed to satisfy:
A⊗ B ⊗ C B ⊗ C ⊗ A
B ⊗ A⊗ C
cA,B⊗C
cA,B⊗1C 1B⊗cA,C
B C A
A B C
=
B ⊗ C A
A B ⊗ C
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Mirror image axiom
A⊗ B ⊗ C C ⊗ A⊗ B
A⊗ C ⊗ B
cA×B,C
1A⊗cB,C cA,C⊗1B
C A B
A B C
=
C A⊗ B
A⊗ B C
This completes the definition of a braided monoidal category.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
The Yang-Baxter equation
We will show that the Yang-Baxter equation is true in a braidedmonoidal category. Remember that this means we have toshow:
C B A
A B C
=
C B A
A B C
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Proof of the Yang-Baxter equation
C B A
A B C
=
cB,C
C ⊗ B A
A B ⊗ C
= cB,C
C ⊗ B A
A B ⊗ C
=
C B A
A B C
Showing that the Yang-Baxter equation is true in a braidedmonoidal category uses both axioms and naturality.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Slogan, revisited
Recall the slogan: modules over a quasitriangular bialgebra area braided category.
We wish to impose properties on a bialgebra H that will makethe module category into a braided modular category.
Thus we will arrive at the notion of a quasitriangular or braidedHopf algebra (QTHA), introduced by Drinfeld in his 1986 ICMlecture.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Quasitriangular Hopf algebras
Let R be an element of H ⊗ H, which we assume to beinvertible. If U and V are H-modules, consider the mapτR : U ⊗V → V ⊗U. (τ is always the flip map x ⊗ y → y ⊗ x ofany tensor product U ⊗ V .)
PropositionSuppose that for x ∈ H we have
R∆(x)R−1 = τ∆(x).
Then for all U,V, the map cU,V : U ⊗ V → V ⊗ U defined bycu,v (u ⊗ v) = τR(u ⊗ v) is an H-module homomorphism.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Proof
We are assuming R∆(x)R−1 = τ∆(x) which we rewrite
R∆(x) =(τ(∆(x))
)R.
We will writeR = R(1) ⊗ R(2),
where we have omitted an implicit summation. Combining thiswith ∆(x) = x(1) ⊗ x(2) (Sweedler notation)
R(1)x(1) ⊗ R(2)x(2) = x(2)R(1) ⊗ x(1)R(2)
in other words
R(1)x(1) ⊗ R(2)x(2) = x(2)R(1) ⊗ x(1)R(2). (1)
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Proof (continued)
Now suppose u ⊗ v ∈ U ⊗ V . We need to show for x ∈ H:
τR(x(u ⊗ v)) = xτR(u ⊗ v)
in other words
τ(R(1)x(1)u ⊗ R(2)x(2)v) = (x(1) ⊗ x(2))τ(R(1)u ⊗ R(2)v)
orR(2)x(2)v ⊗ R(1)x(1)u = x(1)R(2)v ⊗ x(2)R(1)u
This follows from (1) by applying the map
a⊗ b 7→ bv ⊗ au
and we are done.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Some notation
Remember that R ∈ H ⊗ H. We have written
R = R(1) ⊗ R(2),
and this is shorthand for a sum, say
R =N∑
i=1
(R′i )⊗ (R′′i ),
where we are writting R(1) instead of R′i and R(2) instead of R′′i .We will consider some elements of H ⊗ H ⊗ H,
R12 = R(1) ⊗ R(2) ⊗ 1R, R13 = R(1) ⊗ 1R ⊗ R(2),
R23 = 1R ⊗ R(1) ⊗ R(2).
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Quasitriangular Hopf algebras
Definition (Drinfeld)A quasitriangular or braided Hopf algebra is a Hopf algebra Hwith R ∈ H ⊗ H such that
R∆(x)R−1 = τ∆(x)
and(∆⊗ 1)R = R13R23, (1⊗∆)R = R13R12.
The element R is called the universal R-matrix. We understandthe significance of the first condition: it means that if U,V areH-modules, then cU,V : U ⊗ V → V ⊗ U defined by(u, v) 7→ τR(u ⊗ v) is an H-module homomorphism. Whatabout the other properties?
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Braided category axioms and quasitriangularity
We will show that the axiom
(1⊗∆)R = R13R12
is equivalent to the axiom
A⊗ B ⊗ C B ⊗ C ⊗ A
B ⊗ A⊗ C
cA,B⊗C
cA,B⊗1C 1B⊗cA,C
First let us argue that the top arrow cA,B⊗C is
cA,B⊗C(a,b, c) = θ((1⊗∆)R)(a⊗ b ⊗ c),
whereθ(a⊗ b ⊗ c) = b ⊗ c ⊗ a.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Braided category axiom continued
From the previous slide, we are checking:
cA,B⊗C(a,b, c) = θ((1⊗∆)R)(a⊗ b ⊗ c), (2)
whereθ(a⊗ b ⊗ c) = b ⊗ c ⊗ a.
Indeed let us treat d = b ⊗ c as a unit and remember thedefinition of cA,D, with D = B ⊗ C.
cA,B⊗C(a⊗ d) = τR(a⊗ d) = τ(R(1)a⊗ R(2)d , )
and θ is just the map τ in this setting. Now since themultiplication of Hopf elements on a tensor product of modulesis through the tensor product,
R(2)d = ∆(R(2))(b ⊗ c).
Thus we obtain (2).
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Braided category axiom continued
Now we want to prove:
A⊗ B ⊗ C B ⊗ C ⊗ A
B ⊗ A⊗ C
θ((1⊗∆)(R))
(τ⊗1C)R12 (1B⊗τ)R23
We need to show that
θ((1⊗∆)R) = (1B ⊗ τ)R12(τ ⊗ 1C)R12.
We have(τ ⊗ 1C)R23(τ ⊗ 1C) = R13
so because (1B ⊗ τ)(τ ⊗ 1C) = θ:
(1B ⊗ τ)R23(τ ⊗ 1C)R12 = (1B ⊗ τ)(τ ⊗ 1C)R13R23 = θR13R23
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Braided category axiom, concluded
Now using the assumption from the definition of a QTHA:
(1⊗∆)R = R13R23,
the commutativity of
A⊗ B ⊗ C B ⊗ C ⊗ A
B ⊗ A⊗ C
θ((1⊗∆)(R))
(τ⊗1C)R12 (1B⊗τ)R23
follows and we have proved one of the braided categoryaxioms.That the mirror image axiom and naturality are satisfied are leftto the exercises.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Example of a QTHA
Let G be the finite cyclic group of order n:
〈g|gn = 1〉
Let H = C[G] be the group algebra and let q = e2πi/n. It is aHopf algebra with comultiplication
∆(ga) = ga ⊗ ga.
DefineR =
1n
∑a,b
q−abga ⊗ gb ∈ H ⊗ H.
(Sum is over a,b mod n). This is invertible with
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
R is invertible
R =1n
∑a,b
q−abga ⊗ gb ∈ H ⊗ H.
R−1 =1n
∑a,b
qabga ⊗ gb.
To see this, the product of these two elements is
1n2
∑a,b,c,d
q−ab+cdga+c⊗gb+d =1n2
∑t ,u
∑a,b
q−ab+(t−a)(u−b)
gt⊗gu.
The inner sum is
qtu∑a,b
q−tb−ua =
{n2 if t = u = 0,0 otherwise.
From this it follows that R−1 is indeed an inverse to R.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Cyclic group QTHA
TheoremH is quasitriangular with universal R-matrix R.
Proof: The axiom R∆(h)R−1 = τ∆(h) for h ∈ H is trival sinceH is both commutative and cocommutative. We have
R13R12 =1n2
∑a,b
q−abga ⊗ 1⊗ gb
∑c,d
q−cdgc ⊗ gd ⊗ 1
=
1n2
∑a,b,c,d
q−ab−cdga+c ⊗ gd ⊗ gb
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Proof, concluded
This equals
1n2
∑t ,b,d
(∑a
q−ab−(t−a)d
)gt ⊗ gd ⊗ gb.
The inner sum is zero unless b = d , so this equals
1n
∑t ,b
(∑a
q−tb
)gt ⊗ gb ⊗ gb = (1⊗∆)R.
This proves R13R12 = (1⊗∆)R and R13R23 = (∆⊗ 1)R issimilar.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Quantized enveloping algebras
We have mentioned in Lecture 1 that there are two classes ofHopf algebras that may be derived from a Lie group: theenveloping algebra, and the affine algebra or coordinate ring.Both admit quantum deformations.
A particular subtle point is the existence of the universalR-matrix. If g is a complex semisimple Lie algebra andH = Uq(g) is the quantized enveloping algebra, the R-matrixdoes not live in H itself but in a completion. (So strictlyspeaking, H does not satisfy the definition of a QTHA.)
Morally, however Uq(g) is a QTHA. Moreover if q is a root ofunity Uq(g) has a finite-dimensional quotient that is strictlyquasitriangular.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Uq(sl2)
Recall that the Lie algebra sl2 is 3-dimensional with basis
E =
(0 10 0
), F =
(0 01 0
), H =
(1 00 −1
)with bracket operations
[E ,F ] = H, [H,E ] = 2E , [H,F ] = −2F .
To construct the quantized enveloping algebra we replace H byan invertible element K that you can think of as
K =
(q
q−1
)so that
KEK−1 = q2E , KFK−1 = q−2F EF − FE =K − K−1
q − q−1 .
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
The comultiplication
So we let Uq(sl2) be the algebra with generators K ,K−1,E andF subject to the above relations. It is not possible to naively setq = 1 and recover U(sl2) but a more careful procedure willproduce this result. This is done in Kassel’s book.
There is an algebra homomorphism ∆ : H → H ⊗ H defined by∆(K ) = K ⊗ K ,
∆(E) = E ⊗ K + 1⊗ E , ∆(F ) = F ⊗ 1 + K−1 ⊗ F ,
Indeed, it is not hard to check that the expressions on the rightsatisfy the same relations as E ,F ,K . There is a counitε : K → C defined by ε(K ) = 1, ε(E) = ε(F ) = 0 and antipodeS : K → K such that S(K ) = K−1, S(E) = −E and S(F ) = −F .So H is a Hopf algebra.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
At a root of unity
We refer to Majid and Kassel for the following facts.
Let q = e2πi/n where we assume that n is odd. Then En, F n
and K n are central, and quotienting by them produces thefinite-dimensional Hopf algebra uq(sl2). It is quasitriangular withuniversal R-matrix
R =1n
n−1∑a,b=0
q−2abK a ⊗ K b
n−1∑r=0
(q − q−1)r
[r ]q−2!E r ⊗ F r
where
r ]q−2! =r∏
t=1
[t ]q−2 , [r ]q−2 =1− q−2r
1− q−2 .
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Knots and links
A knot is a smooth simple closed curve in S3. A link is a finiteunion of disjoint smooth simple closed curves.
Knot Theory (Wikipedia)
Knotinfo web page
The Knot Atlas
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Wild knots
We only consider knots and links that are equivalent by ambientisotopy to a smooth curve or (equivalently) a finite union ofsegments, to avoid wild knots like this one:
Wild knots (Wikipedia Link)
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Reidemeister moves
Knots are typically studied by projecting them onto the plane. Aknot in 3-space is projected onto R2 with the crossings markedto show which strand is over and which is under.
An issue then is to determine when two knots are equivalent byan ambient isotopy (i.e. isotopy of the ambient S3).
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Reidemeister moves of Type I
If the knots are represented by their two-dimensionalprojections, a necessary and sufficient condition is that theseprojections be related by a sequence of Reidemeister moves.There are three kinds. A Reidemeister move of Type I undoes atwist:
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Reidemeister moves of Type II
A Reidemeister move of type II changes:
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Reidemeister moves of Type III
A Reidemeister move of type III changes:
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Difficulty with Type I
We note that Reidemeister moves of Types II and III haveanalogs in braided rigid categories. However a Reidemeistermove of Type I does not. In a braided rigid category, the twistthat we drew is a morphism V ∗∗ → V :
V
V ∗∗
V ∗
Since this morphism doesn’t map to the same object, we do notexpect to be able to straighten it.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Framed knots and links
Instead of working with knots, we wish to work with framed links(“ribbons”) which can also be projected to the plane, and whichare then equivalent when they are related by Reidemeistermoves of Types II and III, but not of Type I.
A framed knot is not just a strand, but a strand with a givennormal vector field. If we fatten it up in the direction of thenormal vector field, it becomes a ribbon, which can twist inspace.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Twisting
If we double the strand in order to visualize how it will behave in3 dimensions, the red strand passes first over, then under theblue strand. When we pull it straight, it gets a double twist.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Evaluation and coevaluation of tensor products
We proved in Lecture 2 that the evaluation and coevaluationcharacterize the dual uniquely.
V
V
V ∗
V
V
V ∗
V ∗
V
V ∗
V ∗
We can use this to identify evV when V = U ⊗W is a tensorproduct.
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Evaluation and coevaluation of tensor products
U W
U W
U∗
W ∗ =
U W
U W
Uniqueness of the dual implies
evU⊗W = evW (1W∗ ⊗ evU ⊗1W ),
coevU⊗W = (1U ⊗ coev W ⊗ 1U∗) coevU
(Other axiom can be diagrammed similarly.)
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
A relation between cV ,W and cV∗,W
We will prove that
(evV ⊗1W )(1V∗ ⊗ cW ,V ) = (1W ⊗ evV )(c−1W ,V∗ ⊗ 1V )
W
V ∗ W V
W
V ∗ W V
The figures are isotopic, so this is expected, but how to proveit?
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Proof
W
V ∗ W V
W
V ∗ W V
What just happened?
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
We used the braided category axiom and naturality
W ⊗ V ∗ ⊗ V V ∗ ⊗ V ⊗W K ⊗W = W
V ∗ ⊗W ⊗ V
cW ,V∗⊗V
cW ,V∗⊗1V
evV
1∗V⊗cW ,V
K W
W V ∗ V
=
K W
W V ∗ ⊗ V
=
K W
W V ∗ ⊗ V
Braided categories Quasitriangular Hopf algebras Quantized enveloping algebras Reidemeister moves Braiding in a rigid category
Exercises
Exercise 1. In the proof that the category of modules for aQTHA is a braided modular category, we left naturality and themirror image axiom to the reader. Check these.
Exercise 2. Prove that
cV∗,W = (evV ⊗1W ⊗ 1V∗)(1V∗ ⊗ c−1V ,W )(1V∗ ⊗ 1W ⊗ coevV ).
W V ∗
WV ∗
=
V ∗
V
W
V ∗ W
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