Table Of ContentHopf algebras, quantum groups and topological
field theory
Winter term 2014/15
Christoph Schweigert
Hamburg University
Department of Mathematics
Section Algebra and Number Theory
and
Center for Mathematical Physics
(asof:16.5.2015)
Contents
1 Introduction 1
1.1 Braided vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Hopf algebras and their representation categories 4
2.1 Algebras and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Coalgebras and comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Tensor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 Examples of Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Finite-dimensional Hopf algebras 54
3.1 Hopf modules and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Integrals and semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Powers of the antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 Quasi-triangular Hopf algebras and braided categories 84
4.1 Interlude: topological field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Braidings and quasi-triangular bialgebras . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Interlude: Yang-Baxter equations and integrable lattice models . . . . . . . . . . 98
4.4 The square of the antipode of a quasi-triangular Hopf algebra . . . . . . . . . . 101
4.5 Yetter-Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
i
5 Topological field theories and quantum codes 118
5.1 Spherical Hopf algebras and spherical categories . . . . . . . . . . . . . . . . . . 118
5.2 Tanaka-Krein reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3 Knots and links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4 Topological field theories of Turaev-Viro type . . . . . . . . . . . . . . . . . . . 134
5.5 Quantum codes and Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5.1 Classical codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5.2 Classical gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.5.3 Quantum computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.5.4 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.5.5 Quantum codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.6 Topological quantum computing and Turaev-Viro models . . . . . . . . . 145
5.6 Modular tensor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.7 Topological field theories of Reshetikhin-Turaev type and invariants of 3-
manifolds and knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A Facts from linear algebra 153
A.1 Free vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.2 Tensor products of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B Glossary German-English 157
Literature:
Some of the Literature I used to prepare the course:
S. Dascalescu, C. Nastasescu, S. Raianu, Hopf Algebras. An Introduction.
Monographs and Textbooks in Pure and Applied Mathematics 235, Marcel-Dekker, New-York,
2001.
C. Kassel, Quantum Groups.
Graduate Texts in Mathematics 155, Springer, Berlin, 1995.
C. Kassel, M. Rosso, Vl. Turaev: Quantum groups and knot invariants.
Panoramas et Synth`eses, Soc. Math. de France, Paris, 1993
S. Montgomery, Hopf algebras and their actions on rings,
CMBS Reg. Conf. Ser. In Math. 82, Am. Math. Soc., Providence, 1993.
Hans-Ju¨rgen Schneider, Lectures on Hopf algebras,
Notes by Sonia Natale. Trabajos de Matema´tica 31/95, FaMAF, 1995.
http://www.famaf.unc.edu.ar/ andrus/papers/Schn1.pdf
The current version of these notes can be found under
http://www.math.uni-hamburg.de/home/schweigert/ws12/hskript.pdf
as a pdf file.
Please send comments and corrections to [email protected]!
These notes are based on lectures delivered at the University of Hamburg in the summer
term 2004 and in the fall terms 2012/13 and 2014/15. I would like to thank Ms. Dorothea
Glasenapp for much help in creating a first draft of these notes and Ms. Natalia Potylitsina-
Kube for help with the pictures. Dr. Efrossini Tsouchnika has provided many improvements to
thefirstversionofthesenotes,inparticularconcerningweakHopfalgebras.Iamalsogratefulto
Vincent Koppen, Svea Mierach, Daniel Nett, Ana Ros Camacho and Lukas Woike for comments
on the manuscript.
ii
1 Introduction
1.1 Braided vector spaces
Let us study the following ad hoc problem:
Definition 1.1.1
Let K be a field. A braided vector space is a K-vector space V, together with an invertible
K-linear map
c : V V V V
⊗ → ⊗
which obeys the equation
(c id ) (id c) (c id ) = (id c) (c id ) (id c)
V V V V V V
⊗ ◦ ⊗ ◦ ⊗ ⊗ ◦ ⊗ ◦ ⊗
in End(V V V).
⊗ ⊗
Remark 1.1.2.
Let (vi)i∈I be a K-basis of V. This allows us to describe c ∈ End(V ⊗V) by a family (ckijl)i,j,k,l∈I
of scalars:
c(v v ) = cklv v .
i ⊗ j ij k ⊗ l
k,l
X
If c is invertible, then c describes a braided vector space, if and only if the following equation
holds:
cpqcynclm = cqrclycmn for all l,m,n,i,j,k I.
ij qk py jk iq yr ∈
p,q,y y,q,r
X X
This is a complicated set of non-linear equations, called the Yang-Baxter equation. In this
lecture, we will see how to find solutions to this equation (and why this is an interesting
problem at all).
Examples 1.1.3.
(i) For any K-vector space V denote by
τ : V V V V
V,V
⊗ → ⊗
v v v v
1 2 2 1
⊗ 7→ ⊗
the map that switches the two copies of V. The pair (V,τ) is a braided vector space, since
the following relation holds in the symmetric group S for transpositions:
3
τ τ τ = τ τ τ .
12 23 12 23 12 23
(ii) Let V be finite-dimensional with ordered basis (e1,...,en). We choose some q K× and
∈
define c End(V V), by
∈ ⊗
q e e if i = j
i i
⊗
c(e e ) = e e if i < j
i j j i
⊗ ⊗
e e +(q q 1)e e if i > j.
j i − i j
⊗ − ⊗
For n = dim V = 2, the vector space V V has the basis (e e ,e e ,e e ,e e )
K ⊗ 1⊗ 1 2⊗ 2 1⊗ 2 2⊗ 1
which leads to the following matrix representation for c:
q 0 0 0
0 q 0 0
.
0 0 0 1
0 0 1 q q 1
−
−
1
Thereadershouldcheckbydirectcalculationthatthepair(V,c)isabraidedvectorspace.
Moreover, we have
(c qid )(c+q 1id ) = 0.
V V − V V
− ⊗ ⊗
For q = 1, one recovers example (i). For this reason, example (ii) is called a one-parameter
deformation of example (i).
1.2 Braid groups
Definition 1.2.1
Fix an integer n 3. The braid group B on n strands is the group with n 1 generators
n
≥ −
σ ...σ and relations
1 n 1
−
σ σ = σ σ for i j > 1.
i j j i
| − |
σ σ σ = σ σ σ for 1 i n 2
i i+1 i i+1 i i+1
≤ ≤ −
We define for n = 2 the braid group B as the free group with one generator and we let
2
B = B = 1 be the trivial group.
0 1
{ }
Remarks 1.2.2.
(i) The following pictures explain the name braid group:
σ =
i
... ...
1 2 i i+1 n
σ σ = = σ σ
j i i j
... ... ...
1 2 i i+1 j j+1 n
σ σ σ = = = σ σ σ
1 2 1 2 1 2
(ii) There is a canonical surjection from the braid group to the symmetric group:
π : B S
n n
→
σ τ .
i i,i+1
7→
There is an important difference between the symmetric group S and the braid group
n
B : in the symmetric group S the relation τ2 = id holds. In contrast to the symmetric
n n i,i+1
group, the braid group is an infinite group without any non-trivial torsion elements, i.e.
without elements of finite order.
Let (V,c) be a braided vector space. For 1 i n 1, define an automorphism of V n by
⊗
≤ ≤ −
c id for i = 1
⊗ V⊗(n−2)
c := id c id for 1 < i < n 1
i V⊗(i−1) ⊗ ⊗ V⊗(n−i−1) −
id c for i = n 1.
V⊗(n−2) ⊗ −
We deduce from the axioms of a braided vector space that this defines a linear representation
of the braid group B on the vector space V n:
n ⊗
2
Proposition 1.2.3.
Let (V,c) with c Aut (V V) be a braided vector space. We have then for any n > 0 a
∈ ⊗
unique homomorphism of groups
ρc : B Aut (V n)
n n → ⊗
σ c for i = 1,2,...n 1.
i i
7→ −
Proof.
The relation c c = c c for i j 2 holds, since the linear maps c and c act on different
i j j i i j
| − | ≥
copies of the tensor product. The relation c c c = c c c is part of the axioms of a
i i+1 i i+1 i i+1
braided vector space in definition 1.1.1. 2
Let us explain why the braid group is interesting: consider the subset Yn Cn = C C
⊂ ×∙∙∙×
consisting of all n-tuples (z1,...,zn) Cn of pairwise distinct points, i.e. such that
∈
z = z for i = j.
i j
6 6
The symmetric group S acts on Y by permutation of entries. The orbit space X = Y /S
n n n n n
is called the configuration space of n different points in the complex plane C. Fix the point
p = (1,2,...n) Y and the quotient topology on X .
n n
∈
Theorem 1.2.4 (Artin).
The fundamental group π (X ,p) of the configuration space X is isomorphic to the braid
1 n n
group B .
n
Proof.
We only give a group homomorphism
B π (X ,p) .
n 1 n
→
We assign to the element σ B the continuous path in the configuration space X described
i n n
∈
by the map
f = (f1,...,fn) : [0,1] Cn
→
given by
f (s) = j for j = i and j = i+1
j
6 6
1
f (s) = (2i+1 eiπs)
i
2 −
1
f (s) = (2i+1+eiπs)
i+1
2
i i+1 s = 1
s = 0
i i+1
Since we identified points, this describes a closed path in the configuration space X . Denote
n
the class of f in the fundamental group π (X ,p) by σˆ . One verifies that the classes σˆ obey
1 n i i
the relations of the braid group. Hence there is a unique homomorphism
B π (X,p).
n 1
→
3
We omit in these lectures the proof that the homomorphism is even an isomorphism.
2
In physics, the braid group appears in the description of (quasi-)particles in low-dimensional
quantum field theories. In this case, more general statistics than Bose or Fermi statistics is
possible.
For the sake of completeness, we finally present
Definition 1.2.5
(i) A braid with n strands is a continuous embedding of n closed intervals into C [0,1]
×
whose image L has the following properties:
f
(i) The boundary of L is the set 1,2,...n 0,1
f
{ }×{ }
(ii) For any s [0,1], the intersection Lf (C s ) contains precisely n different points.
∈ ∩ ×{ }
(ii) Braids can be concatenated.
(iii) There is an equivalence relation on the set of braids, called isotopy such that the set
of equivalence classes with a composition derived from the concatenation of braids is
isomorphic to the braid group.
One of our goals is to present a general mathematical framework in which representations of
the braid group can be produced. This framework will incidentally allow to describe a variety
of physical phenomena:
Universality classes of low-dimensional gapped systems.
•
Candidates for implementations of quantum computing.
•
Quantum groups also describe symmetries in a variety of integrable systems, including in
•
particular sectors of Yang-Mills theories.
Italsoproducesrepresentationtheoreticstructuresthatariseinmanyfieldsofmathematics,
ranging from algebraic topology to number theory. In particular, it is clear that when one closes
a braid, one obtains a knot, hence there is a relation to knot theory.
2 Hopf algebras and their representation categories
2.1 Algebras and modules
Definition 2.1.1
1. Let K be a field. A unital K-algebra is a pair (A,μ) consisting of a K-vector space A and
a K-linear map
μ : A A A
⊗ →
such that there is a K-linear map
η : K A ,
→
called the unit, such that
4
(a) μ (μ id ) = μ (id μ) (associativity)
A A
◦ ⊗ ◦ ⊗
(b) μ (η id ) = μ (id η) = id (unitality)
A A A
◦ ⊗ ◦ ⊗
In the first identity, the identification (A A) A = A (A A) of tensor products
∼
⊗ ⊗ ⊗ ⊗
of vector spaces is tacitly understood. Similarly, in the second equation, we identify the
tensor products K A = A = A K. We also write a b := μ(a,b).
∼ ∼
⊗ ⊗ ∙
2. A morphism of algebras (A,μ,η) (A0,μ0,η0) is a K-linear map
→
ϕ : A A ,
0
→
such that
ϕ μ = μ (ϕ ϕ) and ϕ η = η .
0 0
◦ ◦ ⊗ ◦
3. Consider again the flip map
τ : A A A A
A,A
⊗ → ⊗
u v v u
⊗ 7→ ⊗
The opposite algebra Aopp is the triple (A,μopp = μ τ ,η). Thus a b = b a.
A,A opp
◦ ∙ ∙
4. An algebra is called commutative, if μopp = μ holds, i.e. if a b = b a for all a,b A.
∙ ∙ ∈
Examples 2.1.2.
1. The unit η is unique, if it exists.
2. The ground field K itself is a K-algebra.
3. For any K-vector space M, the vector space End (M) of K-linear endomorphisms of M
K
is a K-algebra. The product is composition of linear maps.
4. Let K be a field and G a group. Denote by K[G] the vector space freely generated by G.
It has a basis labelled by elements of G which we denote by a slight abuse of notation by
(g) . The multiplication on basis elements g h = gh is inherited from the multiplication
g G
∈ ∙
of G. It is thus associative, and the neutral element e G provides a unit.
∈
We introduce a graphical calculus in which associativity reads
=
Our convention is to read such a diagram from below to above. Lines here represent the
algebraA,trivalentverticeswithtwoingoingandoneoutgoinglinethemultiplicationmorphism
μ. The juxtaposition of lines represents the tensor product. We have identified again the tensor
products (A A) A = A (A A).
∼
⊗ ⊗ ⊗ ⊗
5
Similarly, we represent unitality by = =
where we identified again the tensor products K A = A = A K. Invisible lines denote the
∼ ∼
⊗ ⊗
ground field K. Note that we have required that the unit element 1A := η(1K) ∈ A is both a
left and a right unit element. If it exists, such an element is unique.
ϕ
=
Amorphismϕofunitalalgebrasobeys
ϕ ϕ
ϕ = = η
and
Alternatively, we can characterize associativity by the following commutative diagram
μ id
A A A ⊗ //A A
⊗ ⊗ ⊗
id μ μ
⊗
(cid:15)(cid:15) (cid:15)(cid:15)
A A // A
⊗ μ
while unitality reads
η id id η
K A ⊗ //A A oo ⊗ A K
⊗ ⊗ ⊗
μ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
A A A
Examples 2.1.3.
1. We give another important example of a K-algebra: let V be a K-vector space. The
tensor algebra over V is the associative unital K-algebra
T(V) = V r.
⊗
r 0
M≥
with the tensor product as multiplication:
(v v v ) (w w ) := v v w w .
1 2 r 1 t 1 r 1 t
⊗ ⊗∙∙∙⊗ ∙ ⊗∙∙∙⊗ ⊗∙∙∙⊗ ⊗ ⊗∙∙∙⊗
The tensor algebra is a Z+-graded algebra: with the homogeneous component T(r) := V⊗r
we have
T(r) T(s) T(r+s) .
∙ ⊂
6
The tensor algebra is infinite-dimensional, even if V is finite-dimensional. In this case,
obviously
dimT(r) = dimV r = (dimV)r .
⊗
On the homogenous subspace V r, it carries an action of the symmetric group S .
⊗ r
2. Denote by I (V) the two-sided ideal of T(V) that is generated by all elements of the form
+
x y y x with x,y V. The quotient
⊗ − ⊗ ∈
S(V) := T(V)/I (V)
+
with its natural algebra structure is called the symmetric algebra over V. The symmetric
algebra is a Z+-graded algebra, as well. It is infinite-dimensional, even if V is finite-
dimensional.
3. Similarly, denote by I (V) the two-sided ideal of T(V) that is generated by all elements
−
of the form x y +y x with x,y V. The quotient
⊗ ⊗ ∈
Λ(V) := T(V)/I (V)
−
with its natural algebra structure is called the alternating algebra or exterior algebra
over V. The alternating algebra is a Z+-graded algebra, as well. If V is finite-dimensional,
n := dimV, it is finite-dimensional. The dimension of the homogeneous component is
n
dimΛr(V) =
r
(cid:18) (cid:19)
A central notion for this lecture is the one of a module:
Definition 2.1.4
Let A be a K algebra. A left A-module is a pair (M,ρ), consisting of a K-vector space M and
a map of K-algebras
ρ : A End (M) .
→ K
Remark 2.1.5.
1. We also write
a.m := ρ(a)m for all a A and m M
∈ ∈
and thus obtain a K-linear map which by abuse of notation we also denote by ρ:
ρ : A M M
⊗ →
a m a.m
⊗ 7→
such that for all a,b A and m,n M and λ,μ K the following identities hold:
∈ ∈ ∈
a.(λm+μn) = λ(a.m)+μ(a.n)
(λa+μb).m = λ(a.m)+μ(b.m)
(a b).m = a.(b.m)
∙
1.m = m
7
(The first two lines just express that ρ is K-bilinear.) For the properties of this map, one
can again use a graphical representation and write down the two commuting diagrams:
A A M μ⊗idM //A M
⊗ ⊗ ⊗
id ρ ρ
⊗
(cid:15)(cid:15) (cid:15)(cid:15)
A M //A
⊗ ρ
while unitality reads
K Mη⊗idM//A M ioodM⊗ηM K
⊗ ⊗ ⊗
ρ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
M M M
2. A right A-module is a left Aopp-module (M,ρ) with ρ : Aopp End(M). We write
→
m.a := ρ(a)m and find the relations:
(λm+μn).a = λ(m.a)+μ(n.a)
m.(λa+μb) = λ(m.a+μ(m.b)
m.(a b) = (m.a).b
∙
m.1 = m
for all a,b A and λ,μ K and m,n M. This explains the word “right module”. This
∈ ∈ ∈
also becomes evident in the graphical notation.
3. To give a module
ρ : K[G] End(M)
→
over a group algebra K[G], it is sufficient to specify the algebra morphism ρ on the basis
(g)g G ofK[G].Thisamountstogivingagrouphomomorphismintotheinvertible K-linear
∈
endomorphisms:
ρ : G GL(M) := ϕ End (M),ϕ invertible .
G → { ∈ K }
The pair (M,ρ ) is called a representation of the group G.
G
Remarks 2.1.6.
1. Any K-vector space M carries a representation of its automorphism group GL(V) by
ρ = id . This representation is called the defining representation of GL(V).
GL(V)
2. Any vector space M becomes a representation of any group G by the trivial operation
ρ(g) = id for all g G.
M
∈
3. A representation (M,ρ) of the free abelian group Z amounts to an automorphism A
∈
GL(M), namely A = ρ(1). Then ρ(n) = An.
4. A representation of the cyclic group Z/2Z on a K-vector space V amounts to an auto-
morphism A : V V such that A2 = id .
V
→
If charK = 2, V is the direct sum of eigenspaces of A to the eigenvalues 1,
6 ±
V = V+ V ,
−
⊕
8
