Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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the generalized polylogarithms, which play a

prominent role these days ranging from applied

particle physics to recent developments in

mathematics.

To determine the Hochschild 1-cocycles of some

Feynman graph Hopf algeb ra H, one determines

first the primitives graphs  of the Hopf algebra,

which, by definition, fulfill the cond ition

ðÞ¼  I þ I   ½61

Using the pre-Lie product above, one then deter-

mines the maps



: H!H

lin

½62

such that



ðhÞ¼B



ðhÞI þ id  B



ðÞðhÞ½63

where B



(h) =



n(, h, ). The coefficients

n(, h, ) are closely related to the section coeffi-

cients noted earlier.

Using the definition of the Bogoliubov map



, this

immediately shows that



ðB



ðhÞÞ ¼





ðhÞ½64

which proves locality of counter-terms upon recog-

nizing that B



increases the augmentation degree.

Here, the insertion of the functions for the subgraph

is achieved using the relevant gluin g data of [36].

To recover the quantum equation of motions from

the Hochschild cohomology, one proves that



¼ 1 þ



SymðÞ



ðX



Þ½65

where



e2

½1

int

v2

½0



½66

has the required solution. Upon application of the

Feynman rules, the maps B



turn into the integral

kernels of the usual Dyson–Schwinger equations.

This allows for new nonpe rturbative approaches

which are a current theme of investigation.

Finally, we note that the 1-cocycles introduced

above allow one to determine sub-Hopf algebras of

the form

ðc

Þ¼

Pðfc

gÞ  c

½67

where the c

are defined in eqn [3]. These algebras

do not necessitate the considerations of single

Feynman graphs any longer, but allow one to

establish renormalization directly for the sum of all

graphs at a given loop order. Hence, they establish a

Hopf algebra structure on time-ordered products in

momentum space. For theories with internal sym-

metries, one expects and indeed finds that the

existence of these subalgebras establishes relations

between graphs that are same as the Slavnov–Taylor

identities between the couplings in the Lagrangian.

Outlook

Thanks to the Hopf and Lie algebra structures

described above, quantum field theory has started to

reveal its internal mathematical structure in recent

years, which connects it to a motivic theory and

arithmetic geometry. Conceptually, quantum field

theory has been the most sophisticated means by

which a physicist can describe the character of the

physical law. We have slowly begun to under-

standing that, in its short–distance singularities, it

encapsulates concepts of matching beauty. We can

indeed expect local point-particle quantum field

theory to remain a major topic of mathematical

physics investigations in the foreseeable future.

See also: Bicrossproduct Hopf Algebras and

Noncommutative Spacetime; Exact Renormalization

Group; Hopf Algebras and q-Deformation Quantum

Groups; Number Theory in Physics; Operads;

Perturbation Theory and Its Techniques;

Renormalization: General Theory; von Neumann

Algebras: Introduction, Modular Theory, and

Classification Theory.

Further Reading

Bloch S, Esnault H, and Kreimer D (2005) On Motives associated

to graph polynomials.

Connes A and Kreimer D (1998) Hopf algebras, renormalization

and noncommutative geometry. Communications in Mathe-

matical Physics 199: 203 (arXiv:hep-th/9808042).

Connes A and Kreimer D (2001) Renormalization in quantum

field theory and the Riemann–Hilbert problem. II: The

beta-function, diffeomorphisms and the renormalization

group. Communications in Mathematical Physics 216: 21 5

(arXiv:hep-th/0003188).

Connes A and Marcolli M (2004) From physics to number theory

via noncommutative geometry. II: Renormalization, the

Riemann–Hilbert correspondence, and motivic Galois theory,

arXiv:hep-th/0411114.

Kreimer D (2003) New mathematical structures in renormaliz-

able quantum field theories. Annals of Physics 303: 179

(arXiv:hep-th/0211136).

Kreimer D (2004) The residues of quantum field theory: numbers

we should know, arXiv:hep-th/0404090.

Kreimer D (2005) Anatomy of a gauge theory.

686 Hopf Algebra Structure of Renormalizable Quantum Field Theory

Hopf Algebras and q-Deformation Quantum Groups

S Majid, Queen Mary, University of London,

London, UK

Introduction

Quantum groups are a remarkable generalization of

conventional groups using an algebraic language by

now quite well known to mathematical physicists.

This language is first and foremost the concept of a

‘‘Hopf algebra.’’ In fact, the axioms of a Hopf

algebra are so attractive from a mathematical point

of view that they were proposed in the 1940s long

before the advent of truly representative examples,

which did not come until the 1980s (from mathe-

matical physics). Until then, they were used mainly

by mathematicians as a way for redoing group

theory and Lie algebra theory in a more uniform

way.

It is remarkable that at least three points of view

lead to the same axioms of a Hopf algebra:

1. Generalized symmetry A generalization of a

usual group algebra or enveloping algebra of a

Lie algebra that can nevertheless act on other

algebraic objects. The structure that controls this

is the ‘‘coproduct’’  : H !H  H, while the

group or Lie structure is encoded in the algebra

H which is typically not changed up to iso-

morphism.  allows H to act on tensor products

and this is needed to define what it means, for

example, for a product A  A !A of an algebra

to be an intertwiner. The usual flip map between

two representations V  W !W  V is not

typically an intertwiner any more, instead that

is provided by an R-matrix solving the Yang–

Baxter equations (YBE).

2. Noncommutative geometry A generalization of

the coordinate algebra of functions on a conven-

tional group to allow noncommutative or ‘‘quan-

tum’’ coordinate algebras. Here the group

structure is encoded in a coproduct  : H !H 

H in a way which would, in the case of functions

on a group, be defined by the group product. It is

typically not changed, the change being in the

algebra.

3. Duality An object that admits observer–

observed duality or Fourier transform. Such a

duality is known for abelian groups, lost for

nonabelian groups but re-emerges for Hopf

algebras. If there is to be an algebra with product

H  H !H, then there should also be a

‘‘coproduct’’  : H !H  H to maintain the

duality symmetry. Then a suitable dual space



is also a Hopf algebra, with the roles of

product and coproduct interchanged.

In line with these main ideas are three known classes

of true quantum groups, and these remain the main

types of example at the time of writing: the q-deformed

enveloping algebras U

(g) of Drinfeld and Jimbo, their

duals as quantizations of the Drinfeld–Sklyanin

Poisson bracket on a simple Lie group (both of these

arising from quantum inverse scattering but also in

the case of C

[SU

]fromC



-algebras) and the

bicrossproduct quantum groups based on Lie group

factorizations (arising from ideas for Planck-scale

physics and quantum gravity). The latter are self-dual

and hence are both generalized symmetries and

noncommutative or quantum geometries at the same

time. The impact of such quantum groups has been

very far reachi ng from a mathematician’s point

of view, spanning revolutions in the theory of knot

and 3-manifold invariants, Poisson geometry, new

directions in noncommutative geometry, to name

some. In physics they are, at the time of writing,

beginning seriously to be applied in a variety of

contexts beyond the original ones, such as in book-

keeping overlapping divergences in general quantum

field theories, quantum computing, and construction

of anyons. This article will mention some of these, but

just as groups have many different roles in physics,

one can expect that quantum groups and variants of

them can and will have diverse roles as well. What

follows is a short overview.

Hopf Algebras and First Examples

The genera l theory works over any field k but (to be

concrete) we write our examples over C; one can

also have examples over, say, the field Z

of two

elements. A Hopf algebra then is:

1. An algebra H with unit which is also a

‘‘coalgebra’’ with counit, that is, there are maps

 : H !H  H,  : H !k obeying:

ð  idÞ ¼ðid  Þ

ð  idÞ ¼ðid  Þ ¼ id

2. ,  should be algeb ra homomorphisms.

3. There should be a map S : H !H called the

antipode or ‘‘linearized inverse’’ obeying

ðid  SÞ ¼ðS  idÞ ¼ 1

Hopf Algebras and q-Deformation Quantum Groups 687

If the third axiom is not obeyed one has a

‘‘quantum semigroup’’ or ‘‘bialgebra.’’ Note also

that S looks nothing like a usual inverse and it is

not, yet it plays the same role. For example, we can

define conjugation or the ‘‘adjoint action’’ of any

Hopf algebra on itself by

ðbÞ¼

ð1Þ

bSa

ð2Þ

; a ¼

ð1Þ

 a

ð2Þ

where we use here the ‘‘Sweedler notation’’ for a a

sum of unspecified pieces in H  H. Moreover, if it

exists, then S is unique and (it can be shown)

S(ab) = (Sb)S(a) for all a, b 2H, just like an inverse.

The self-duality of these axioms is evident from

the first one: a coalgebra is just an algebra with its

product map H  H !H, unit element (viewed as a

map k !H sending 1 to 1) and the associativity and

unity axioms all written backwards. Meanwhile,

the middle axiom means in explicit terms

(ab) = (a)(b), (ab) = (a)(b) for all a, b 2H

and (1) = 1  1, (1) = 1. This may not look self-

dual but it is equivalent to saying that the product

and unit are coalgebra homomorphis ms. Indeed, if

one takes the trouble to write out all the axioms as

commutative diagrams, the set of axioms is invar-

iant under arrow reversal. Such arrow reversal can

also be concretely implemented, for example, by

taking adjoints. Thus, the coproduct dualizes to a

map (H  H)



and since H



 H



 (H  H)



we have a product on the dual H



. If the dual space

is defined correctly, one also has a coproduct by

dualizing the product, etc. One says that two Hopf

algebras H, H

are ‘‘in duality’’ if their maps are

adjoint to each other in such a way.

The role of quantum groups as generalized

symmetries is typified by the following examples.

Thus, let G be a group; then its group algebra CG

defined as a vector space (written here over C) with

basis identified with G and product given by the

group product extended linearly, is a Hopf algebra

with

g ¼ g  g;g ¼ 1; Sg ¼ g

1

; 8g 2G

Likewise, if g is a Lie algebra, then its universal

enveloping algebra U(g) generated by g is a Hopf

algebra with

 ¼   1 þ 1  ;  ¼ 0; S ¼; 8 2 g

The two examples are related if one informally

allows exponentials, then g = e



has coproduct

e



¼ e



¼ e

1þ1

¼ e



 e



using axiom 2 and that   1, 1   commute in the

tensor product algebra.

The coproduct structures are therefore implicit

already in Lie theory and group theory. As for any

Hopf algebra , specifies how the algebra H acts in

a tensor product of two representations. For groups

the tensor product is diagonal (g acts on each copy),

for Lie algebras it is additive (e.g., the addition of

angular momenta). In general, the action of a 2H is

defined as the action of a on the tensor product.

This has far-reaching consequences. For example,

for the product A  A !A of an algebra to be

covariant means that H acting before and after the

product map gives the same answer, similarly for the

unit map where k has the trivial representation

afforded by , that is,

h . ðabÞ¼

ðh

ð1Þ

. aÞðh

ð2Þ

. bÞ; h . 1 ¼ ðhÞ1

for all a, b 2A and h 2H. What that means in the

case of a group is therefore g . (ab) = (g . a)(g . b)or

G acts by automorphisms. What it means for a Lie

algebra is .(ab) = (.a)b þ a(.b), that is, g acts

by derivations. This is how Hopf algebra theory

unifies group theory and Lie algebra theory and

potentially takes us beyond.

In another, dual, point of view, if G is a group

defined by polynomial equations in C

, then the

Hilbert’s ‘‘nullstellensatz’’ in algebraic geometry says

that it corresponds algebraically to a commutative

nilpotent-free algebra with n generators, called its

‘‘coordinate alge bra’’ H = C[G]. The group product

then corresponds to  making C[G] into a Hopf

algebra. If one replaces C by any field, one has an

algebraic group over the field. For example, the

group SL

has coordinate algebra gener-

ated by four functions a, b, c, d where a at matrix

g 2SL

the 1,1 entry of the matrix,

similarly b(g) = g

etc. Then C[SL

] is the commu-

tative algebra generated by a, b, c, d with the relation

ad  bc = 1. A little thought about matrix multi-

plication should convince the reader that









where we have written the operation on each

generator as an array and where matrix multi-

plication is understood (so a = a  a þ b  c, etc.).

The counit and antipode are





d b

ca



One could also let G be a finite group, in which case

the algebra C(G) of (say complex-valued) functions

on it is more obviously a Hopf algebra with

ðaÞðg; hÞ¼a ðghÞ;ðaÞ¼að1Þ; ðSaÞðgÞ¼aðg

1

688 Hopf Algebras and q-Deformation Quantum Groups

for any function a 2C(G). Here we identify C(G) 

C(G) = C(G G) or functions in two variables on

the group. These examples are dually paired with

U(g) in the Lie case and CG in the finite case,

respectively.

In such a coordinate algebra point of view, usual

constructions in group theory appear expressed

backwards with arrows reversed. So an action of

the group appears for such a Hopf algebra H as a

‘‘coaction’’ 

: V !V  H (here a right coaction,

one can similarly have 

a left coaction). It obeys

ð

 idÞ

¼ðid  Þ

; ðid  Þ

¼ id

which are the axioms of an algebra acting written

backwards for the coalgebra of H ‘‘coacting.’’ An

example is the right action of a group on itself which

in the coordinate ring point of view is 

=,that

is, the coproduct viewed as a right coaction. It is the

algebra of H that determines the tensor product of

two coactions, so, for example, A is a coaction

algebra in this sense if 

: A !A  H is a coalgebra

and an algebra homomorphism. Similarly, in this

coordinate point of view, an integral on the group

means a map

: H !k and right invariance trans-

lates into invariance under the right coaction, or

id



 ¼ 1

There is a theorem that such an integration, if it

exists, is unique up to scale. In the finite-dimensional

case it always exists, for any field k. At leas t in this

case, let exp =

 f

for a basis {e

}ofH and {f

}

a dual basis. Then an application of the integral is

Fourier transform H !H



defined by

FðaÞ¼

a  f

with properties that one would expect of Fourier

transform. The inverse is give n similarly the other

way up to a normalization factor and using the

antipode of H. This is one among the many results

from the abstract theory of Hopf algebras, see

Sweedler (1969) and Larson and Radford (1988)

among others.

A given Hopf algebra H does not know which

point of view one is taking on it; the axioms of a

Hopf algebra include and unify both enveloping and

coordinate algebras. So an immediate consequence is

that constructions which are usual in one point of

view give new constructions when the wrong point

of view is taken (put another way, the self-dua lity of

the axioms means that any general theorem has a

second theorem for free, given, if we keep the

interpretation of H fixed, by reversing all arrows in

the original theorem and its proof). Even the

elementary examples above are quite interesting for

physics if taken ‘‘upside down’’ in this way. For

example, if G is nonabelian, then CG is noncom-

mutative, so it cannot be functions on any actual

group. But it is a Hopf algebra, so one could think

of it as being like C(

G), where

G is not a group but

a quantum group defined as C(

G) = CG. The latter

is a well-defined Hopf algebra viewed the wrong

way. So this is an ap plication of noncommutative

geometry to allow nonabelian Fourier transform

F : C(G) !CG. Similarly, U(g) is noncommutative

but one could view it upside down as a quantization

of C[g



] = S(g) (the symmetric alge bra on g). To do

this let us scale the generators of g so that the

relations on U(g) have the form    = [, ]

where  is a deformation parameter. Then the

Poisson bracket that this algebra quantizes

(deforms) is the Kirillov–Kostant one on g



where

{, } = [, ]. Here ,  on the left-hand side are

regarded as functions on g



, while on the right-hand

side we take their Lie bracket and then regard

the result as a function on g



. Examples which

have been used successfully in physics include:

½t; x

¼ix

bicrossproduct model R

1;3





½x

; x

¼i2

ijk

spin space model R





(summation understood over k). In both cases, we

may develop geometry on these algebras using

quantum group methods as if they were coordinates

on a usual space (see Bicrossproduct Hopf Algebras

and Noncommutative Spacetime). They are versions

of R

because the co product which expresses the

addition law on the noncommutative space is the

additive one according to the above. In the second

case, setting the Casimir to the value for a spin j is the

quadratic relation of a ‘‘fuzzy sphere.’’ As algebras,

the latter are just the algebras of (2j þ 1)(2j þ 1)

matrices.

Going the other way, we can take a classical

coordinate ring C[G] and regard it upside down as

some kind of group or enveloping algebra but with

a nonsymmetric . In the finite group case, an

action of C(G) just means a G-grading. Here if an

element v of a vector space has G-valued degree jvj

then a . v = a(jvj)v is the action of a 2 C(G).

Alternatively, this is the same thing as a right

coaction of CG, 

v = v jvj. Thus, the notion of

group representation and group grading are also

unified. This is familiar in physics for abelian

groups (a U(1) action is the same thing as a

Z-grading) but works fine using Hopf algebra

methods for nonabelian groups and beyond.

Hopf Algebras and q-Deformation Quantum Groups 689

Returning to axioms, if one wants to speak of real

forms and unitary representations, this corresponds,

for Hopf algebras, to H a -algebra over C with





ðÞ¼ð  Þ; S ¼ S

1



where  (throughout this article) denotes transposi-

tion of tensor factors. This requires in particular that

S in invertible (which is not assumed for a general

Hopf algebra though it does hold in the finite-

dimensional case and in all examples of interest).

Thus, C[SU

] denotes the above with a certain 

structure whereby the matrix of generators is unitary.

q-Deformation Enveloping Algebras

For a genuinely representative example of a Hopf

algebra, consider, U

(sl

) defined with noncommu-

tative generators and relations, coproduct etc.,

h=2



h=2

¼ q

1



½x

; x



¼

 q

h

q  q

1

x



¼ x



 q

h=2

þ q

h=2

 x



q

h=2

¼ q

h=2

 q

h=2

x



¼ 0;q

h=2

¼ 1



¼q

1



; Sq

h=2

¼ q

h=2

The actual generators here are x



, q

h=2

but the

notation is intended to be suggestive: if h existed and

we took the limit q !1, we would have the usual

enveloping algebra of the Lie algebra sl

. The

quantum group U

(su

) is the same with the

-structure h



= h, x





= x



when q is real (there are

other possibilities).

Two words of warning here. Although some

authors write q = e

h=2

, the parameter q here has little

to do with quantization. In fact, the cases of direct

relevance to physics are q

2i=(2þk)

, where k is the level

of the Wess–Zumino–Witten (WZW) model in which

this quantum group appears as a generalized symme-

try. This quantum group also (first) appeared in the

theory of exactly solvable lattice models, namely the

Ising model with an applied exter nal magnetic field:

q 6¼ 1 is a measure of the resulting nonhomogeneity

of the model. Its origins go further back to the

algebraic Bethe ansatz and the emergence of the YBE

in such models (Baxter 1982). The genera l U

(g)

emerged from this context in Drinfeld (1987) and

Jimbo (1985) and the same rem ark applies (see Affine

Quantum Groups; Yang–Baxter Equations).

The second warning is that at least informally (if

one works with H and allows formal power series

etc.), the algebra here is isomorphic to usual U(sl

that is, it looks deformed but the true deformation is

not here but in the coproduct, which enters into the

tensor product of representations. The latter are

labeled as usual because the algebra is not really

changed, for example, the unitary ones of U

(su

)

are labeled by spin. The spin-

one even looks the

same with x



, h represented by the standard Pauli

matrices. Tensor products of representations start

to look different but their multiplicities are the same

as classically and if V,W are representations then

V  W ﬃ W  V. Because the coproduct above is

not symmetric in its two factors, this isomorphism



V, W

=   R

V, W

has R

V, W

nontrivial. From the

formulas given, the reader can compute that

1=2;1=2

¼ q

1=2

q 000

01q  q

1

00 1 0

00 0 q

in a tensor product basis. For this particular

quantum group, and others like it, one finds that

these ‘‘R-matrices’’ obey the braid relations as a

version of the YBE. As a result, they can and do lead

to knot invariants; the one above leads to the Jones

knot invariant as a polynomial in q. Briefly, one

represents the knot on a plane, assigns R or R

1

each braid crossing and takes a suitable trace (see

The Jones Polynomial).

Since such features hold in any representation,

these matrices are in fact representations of an

invertible element R2H  H provided one allows h

as a generator and formal power series:

R¼q

ðhhÞ=2

ðqq

1

Þef

2

; e ¼ x

h=2

; f ¼ q

h=2



where

ðxÞ¼

m¼0

½m

; ½m

1  q

are the q-exponential and q-intege r, respectively.

Their proper explana tion is in the section ‘‘Braided

group s and qua ntum planes .’’ ThisR is called the

‘‘uni versal R- matrix’’ or quasitrian gular structure

and obeys

 ¼Rð ÞR

1

ð  idÞR ¼ R

; ðid  ÞR ¼ R

and from the axioms of a Hopf algebra, one may

deduce that the YBE

¼R

hold in the algebra. This induces the YBE for

matrices R

V, W

in the representation V  W.Sucha

690 Hopf Algebras and q-Deformation Quantum Groups

Hopf algebra is called ‘‘quasitriangular’’ and its

representations form a braided category (see Braided

and Modular Tensor Categories). Even if R for a

quasitriangular Hopf algebra is defined by a power

series, the R

V, W

in finite-dimensional representa-

tions are typically actual matrices.

Of considerable interest is the special case when q

is a primitive nth root of unity. In this case the

quasitriangular Hopf algebra u

(sl

) has the above

generators but the additional relations

¼ f

¼ 0; g

¼ 1; g ¼ q

which render the algebra generated by e, f , g as n

dimensional. The algebra no longer has a matrix

block decomposition (is not semisimple) and not all

representations descend to it. For example, if n is

odd, then only representations of dimension n

descend. Other than this, one has many of the

features of a classical enveloping algebra now for

this finite-dimensional object. There is evidence that

such objects over C are intimately related to

classical Lie algebras but over a finite field.

Finally, there is a similar theory of U

(g) for all Lie

algebras determined by symmetrizable Cartan matrices

}, including affine ones. Here i, j 2I an indexing set

and a

= 2i  j=i  i 2 {0, 1, 2, ...}fori 6¼ j,where

is a symmetric bilinear form on the root lattice Z[I]

generated by I with i  i a positive even integer. To be

precise, one should also fix a ‘‘root datum’’ in the form

of an inclusion Z[I]  X of the root lattice into a choice

of character lattice X and an inclusion Z[I]  Y of the

coroot lattice (also labeled by I) into the cocharacter

lattice Y (the dual of X). Here the evaluation pairing is

required to restrict to hi, ji= a

if i, j 2I and i is i

viewed in the cocharacter lattice Y.Weletq

= q

ii=2

and require q

6¼ 1 for all i (or one may consider q as an

indeterminate). We have generators e

, f

for i 2I and

invertible g

for a each generator of Y, and the relations

¼ q

ha;ii

; f

¼ q

ha;ii

½e

; f

¼

ii=2

i

 g

ii=2

i

 q

1



1a

r¼0

ð1Þ

1  a



ðe

1a

r

¼ 0

for all i 6¼ j and an identical set for the {f

}. The

coalgebra and antipode are

e

¼ e

 g

ii=2

i

þ 1  e

f

¼ f

 1 þ g

ii=2

i

 f

g

¼ g

 g

;ðg

Þ¼1;ðe

Þ¼ðf

Þ¼0

¼ g

1

; Se

¼e

ii=2

i

; Sf

¼g

ii=2

i

The q-Serre relations are those above involving the

q-binomial coefficients, defined now using the

symmetric q-integers (m)

= (q

 q

m

)=(q  q

1

They have their true explanation as

ðe

1a

ðe

Þ¼0

where Ad is a braided group adjoint action in the

sense of the sect ion ‘‘Brai ded groups and quantum

planes .’’ Notice that while the roo t genera tors are

modeled on the Lie algebra, the Cartan g enerators

are modeled on the torus of an algebraic group,

which contains global information. Thus, the more

precise form of U

(sl

) is the e, f, g form with the

generator g = q

as above, with Z[I] ( X and

Z[I] = Y. Meanwhile U

(psl

) has the square root

of this as generator (what we called q

h=2

before)

with Z[I] = X and Z[I] ( Y where the strict inclu-

sion has 1= 2 in the lattice Z. Note that, in the

complex case, SL

has compact real form SU

while

its quotient, PSL

, has compac t real form SO

,so

these are distinguished at the Hopf algebra level. In

general, the root datum has an associated reductive

algebraic group which is simply connected when

Y = Z[I] and generated by its adjoint representation

when X = Z[I]. The complexified character lattice is

a sublattice of the more familiar Lie algebra weight

lattice and labels representations that extend to the

(algebraic) group. Langlands duality interchanges

the roles of X, Y. These subtleties are lost when we

work over formal power series with q = e

=2

and

Lie-algebra-like Cartan generators.

These objects are mathematically so interesting

that some authors define ‘‘quantum groups’’ as

nothing more than this pa rticular extension of the

theory of Lie algebras, Cartan matrices and root

systems. Amon g the deepest theorems is the exis-

tence of the Lusztig–Kashiwara canonical basis

which is obtained from q = 0 but valid also at

q = 1 (i.e., for classical enveloping algebras) and

which has the remarkable property of inducing bases

coherently across highest-weight representations.

From a physicist’s point of view, however, there

are many other Hopf algeb ras rather more closely

connected with actual quantization. Most often, the

terms quantum group and Hopf algebra are used

interchangeably.

There is similarly a reduced version u

(g). The

simplest of all possible cases, even simpler than

(sl

), is for what one could call u

(1) with a single

generator g and

¼ 1; g ¼ g  g;g ¼ 1; Sg ¼ g

1

n1

a;b¼0

ab

 g

Hopf Algebras and q-Deformation Quantum Groups 691

where q is a primitive nth root of unity. The Hopf

algebra is the same as the group algebra

= C(

) but the R is nontrivial. A representa-

tion means a

-graded space, that is, graded into

degrees 0, 1, ..., n  1. The braiding matrices have

the diagonal form R

, W

= q

on components of

degree a, b, respectively. The braided category

generated in this case is the one where anyons live.

From this point of view, u

(g) gen erate the category

where nonabelian anyons live. Here R

2 (in place of

(hh)=2

) along with an additional e

2 factor as

above gives the quasitriangular structure of u

(sl

The physical model here is the rational conformal

field theory mentioned above with these anyons as

particular bound states. There is a proposal to use

them in the construction of quantum computers.

q-Deformation Coordinate Algebras

From the coordinate algebra point of view, the

corresponding deformation to the one in the last

section is the Hopf algebra C

[SL

] with noncom-

muting generators and relatio ns

ca ¼ qac; ba ¼ qab

db ¼ qbd; dc ¼ qcd

bc ¼ cb; da  ad ¼ðq  q

1

Þbc

ad  q

1

bc ¼ 1

The coalgebra has the same matrix form on the

generators as for C[SL

] and the antipode and

-structure (for C

[SU

]) are



d qb

q

1





Its duality pairing with U

(sl

) is afforded by the

2 2 Pauli-matrix representation of the latter. The

[SU

] Hopf -algebra may be completed to a



-algebra.

One similarly has C

[G] for all semisimple Lie

groups G and their various real forms. From an

axiomatic point of view, such quantum groups are

‘‘coquasitriangular’’ in the sense that there is a map

R: H  H !k such that



ð1Þ

 b

ð1Þ



ð2Þ

ð1Þ

R a

ð2Þ

 b

ð2Þ



for all a, b 2H and

Rðab  cÞ¼

R a  c

ð1Þ



R b  c

ð2Þ



Rða  bcÞ¼

R a

ð1Þ

 c



R a

ð2Þ

 b



for all a, b, c 2H. We also require that R is

invertible in a certain sense. These are just the

arrow reversal of the axioms of a quasitriangular

structure. In general, for the deformation of a linear

algebraic group we will have some n

generators t

now taken to be noncommutative, and with a

matrix form of coalgebra

t

¼ t

 t

;t

¼ 

For the compact real form we will have St

= t

j

Moreover, from the first of the above axioms we

will have among the relations

¼ t

where R

= R(t

 t

) is a matrix R 2M

 M

obeying the YBE. If we take only these quadratic

relations, we have the ‘‘Faddier Reshetikhin Takhta-

jan (FRT) bialgebra’’ A(R) and it can be shown (see

Majid 1995) that R extends to a coquasitriangular

structure R on it. However, in our case we also have

1i

¼R St

 t



¼R t

 St



where

R = ((R

)

1

)

transposition in the second

factor of M

) is called the ‘‘second inverse’’ of R.With

these additional matrices, one may define a q-determi-

nant and antipode relations as well (Majid 1995). One

may also generate a rigid braided monoidal category

and reconstruct a Hopf algebra



A(R) from it. In this

way, the R-matrix plays a role similar to that of the

structure constants of a Lie algebra and can in

principle define the quantum group coordinate alge-

bra. Such R-matrices have been classified in low

dimension and include multiparameter and other

deformations of classical group coordinate algebra as

well as other nonstandard quantum groups.

In the C

[G] examples it is not the coalgebra which

is essentially deformed but the algebra. We already see

this above on the generators but the coproduct of a

product of generators may look different. Nonetheless,

one can identify the vector space that the products

generate with that of C[G] and at least informally with

respect to a deformation parameter express the

product as a power series in the undeformed product

(a -product deformation). For generic values, one still

has a Peter–Weyl decomposition C

[G] =  (V  V



where the sum is over irreducibles corepresentations,

which can be identified with the classical representa-

tions of the algebraic group. One can make the same

decomposition for C[G] and identify the matrix blocks

V  V



in order to find this -product. Also, since this

is a flat deformation, it follows that the commutator at

lowest order defines a Poisson bracket on G,givenby

; t

g¼t

 r

692 Hopf Algebras and q-Deformation Quantum Groups

and this Poisson bracket is compatible with the group

product G G !G as a Poisson map (because the

Hopf algebra coproduct was an algebra map). Here r

is the first order part in the expansion of the

R-matrix. A Lie group equipped with a Poisson

bracket compatible in this way is called a ‘‘Poisson

Lie group.’’ On general functions its Poisson bivector

is generated by the first order part r 2 g  g in the

expansion of R in the q-deformed enveloping

algebra. In place of the YBE obeyed by R,wehave

the ‘‘classical Yang–Baxter equations (CYBE),’’

½r

; r

þ½r

; r

þ½r

; r

¼0

In this way, one may characterize an ‘‘infinitesimal

version’’ of U

(g)as(g, r, ) where  : g !g  g is

the leading part of    and makes the triple into

a quasitriangular ‘‘Lie Bialgebra’’ ( see Classical

r-Matrices, Lie Bialgebras, and Poisson Lie Groups).

Finally, returning to our example, when q is an

nth root of unity, one has the q-Frobenius Hopf

algebra homomorphism

C½SL

,!C

½SL





that is, a classical copy sitting inside the quantum

group. Quotienting by this means adding the

relations

¼ d

¼ 1; b

¼ c

¼ 0

which gives the finite-dimensional reduced quantum

group C

red

[SL

]. Similarly for other C

red

[G]. These

reduced quantum groups provide finite noncommu-

tative geometries having the geometric flavor of the

classical geometry but where geometry and physics

(such as electromagnetic gauge theory modes) are

fully computable.

Self-Dual Quantum Groups

The arrow-reversibility of the axioms of a quantum

group make it possible to search for self-dual

quantum groups or for quantum groups which, if

not self-dual, have a self-dual form. This leads to the

bicrossproduct quantum groups coming from mod-

els of quantum gravity (Majid 1988)(see Bicross-

product Hopf Algebras and Noncommutative

Spacetime).

The context here is that of Figure 1 which shows

how Hopf algebras relate to other objects and to

duality in a representation-theoretic sense. Along the

central axis, we have put self-dual categories or in

physical terms categories admitting Fourier trans-

form. This is clear for abelian Groups where the

dual

G of an abelian group G is also an abelian

group. Below the axis, we have nonabelian groups

which we view as toy models of geometries with

curvature. Every compact Lie group, for example,

has an associated Killing metric. Above the axis, a

nonabelian group dual

G means to construct unitary

representations etc., which we view as toy models of

quantum theory. We have seen that Hopf algebras

are another self-dual category and provide a frame-

work in which both groups and group duals can be

unified (see the section ‘‘Hopf algebras and first

examples’’). Thus, G can be viewed as a coordinate

Hopf algebra C(G)orC[G] in the finite or Lie cases,

and

G as the dual Hopf algebra CG or U(g)asa

definition of the coordinate algebra ‘‘C(

G).’’ Note that

G is not merely the set of representations, as these

alone are not enough to reconstruct the group (e.g.,

both S

and SO

have the same set). We see that Hopf

algebras are a microcosm for the unification of

quantum theory and gravity. Hopf algebra duality

interchanges the role of position and momentum on

the one hand and of quantum and gravitational effects

on the other. A self-dual Hopf algebra has both aspects

unified and interchanged by the self-duality.

One can also ask what the next most general sel f-

dual category of objects is in which to look for more

general unifications. One answer here is the category

whose objects are themselves categories C equipped

with a tensor product (a ‘‘monoidal category’’) and a

monoidal functor to a fixed monoidal category V.

Motivated by the above, a theorem from the 1980s

is that for any such C there is a dual C

‘‘representations in V’’ (Majid 1991a). The dotted

arrows in Figure 1 indicate that this may be a setting

for more ambitious models than those achieved by

Hopf algebras alone. In fact, the C

construction was

one of the ingredients going into the invention of

2-categories a few years later. See also several

articles on TQFT (such as Topological Quantum

Field Theory: Overview; Axiomatic Approach to

Topological Quantum Field Theory; Duality in

Topological Quantum Field Theory).

Monoidal

categories

Hopf

algebras

Abelian

groups

Nonabelian

groups

Group

duals

Riemannian

geometry

Quantum

theory

Figure 1 Role of Hopf algebras along the self-dual axis.

Hopf Algebras and q-Deformation Quantum Groups 693

The simple st self-dual quantum group is C[x]as

the Hopf algebra of polynomial functions on a line

with additive coproduct. This is dually paired with

itself in the form of the enveloping algebra

U(gl

) = C[p] with pairing

; x

i¼ðiÞ



m;n

and similarly for higher-dimensional flat space. In

the case of C[x], a basis is x

and from the above the

dual basis is (ip)

=n!. Hence the canonical element is

exp = e

ixp

so that Hopf algebra Fourier transform

on a suitable completion of these algebras reduces to

usual Fourier transform.

A more nontrivial example (Majid 1988) is given

by the ‘‘Planck-scale Hopf algebra’’ C[x]

C [p]

which has algebra and coalgebra

½p; x¼ihð1  e

x

Þ; x ¼ x  1 þ 1  x

p ¼ p  e

x

þ 1  p;x ¼ p ¼ 0

Sx ¼x; Sp ¼pe

x

The actual generator here should be e

x

rather than

x for an algebraic treatment (otherwise one should

allow power series or use C



-algebras). The dually

paired Hopf algebra has the same form C[p]

C[x],

with new parameters h

= 1=h and 

=h and

quantum group Fourier transform connects the

two. More details and the general construction of

Hopf algebras C[M]

U(g) with dual U(m) C[G]

are in the article on ‘‘bicrossproduct’’ Hopf algebras

(see Bicrossproduct Hopf Algebras and Noncommu-

tative Spacetime). These quantize particles in M

moving under momentum Lie group G with Lie

algebra g and vice versa. The states of one (in a



-algebra context) lie in the algebra of observables of

the other (‘‘observable–state duality’’). The data

required are a matched pair of actions of (G, M)

on each other. Such equations correspond locally to

a factorization of a larger group G ﬄ M but

typically have singularities and other features in

keeping with a toy model of Einstein’s equations.

There are, by the time of writing, many applica-

tions of bicrossproducts beyond the original one,

including a Poincare´ quantum group for the R

1, 3



mentione d in the section ‘‘Hopf algeb ras a nd

first exampl es,’’ with li nks to Planck-sca le physic s.

There is also a bicrossproduct quantum group

C[G



] U(g) canonically associated to any simple

Lie algebra g and related to T-duality. The classi cal

data here are Lie bialgebras and solut ions of the

CYBE as in the section ‘‘q-Deformation coordinate

algebras,’’ however there is no known relation with

the q-deformation Hopf algebras themselves. Finite

group bicrossproducts are also interesting and

examples (but not with both actions nontrivial)

were already in the works of GI Kac in the 1960s.

These constructions also work when the groups

above are themselves Hopf algebras. For example,

any finite-dimensional Hopf algebra H has a

‘‘quantum double’’ D(H) = H ﬄ H

op

, where the

double cross product ﬄ is by mutual coadjoint

actions. The cross-relations between the two sub-

Hopf algebras are

ð1Þ

; a

ð1Þ

ð2Þ

ð1Þ

ð2Þ

; a

ð2Þ

for h 2H and a 2H



. The construction is due to

Drinfeld (1987) while the ﬄ form is due to the

author. Moreover, D(H) is quasitriangular with

R= exp , the canonical element used in the Fourier

transform on H. Its represent ations consist of vector

spaces where H acts and at the same time H

op

acts

or (which makes sense when H is infinite dimen-

sional) where H coacts, in a compatible way. Such

objects are called ‘‘crossed modules’’ because when

H = CG, one has exactly a linearization of the

crossed G-sets of JC Whitehead. They are a special

case of the C

construction mentioned above.

Finally, one can also view the q-deformed linear

spaces on which quantum groups such as U

(g) act

as self-dual Hopf algebras under an additive

coproduct. However, this needs to be as braided

groups or Hopf algebras with braid statistics, see the

next section. The simplest example here is the

‘‘braided line’’ B = C[x] developed not as above but

as a self-dual Hopf algebra with q-statistics. Its

‘‘bosonization’’ gives a self-dual Hopf algebra

)  U

(sl

), and similarly for other U

) 

(g). Perhaps more surprisingly, the quantum

groups U

(g)andC

[G] also both have canonical

braided group versions (a process called ‘‘transmuta-

tion’’) and as such they too are isomorphic. This

isomorphism extends the linear isomorphism g !g



afforded by the Killing form of any semisimple Lie

algebra. In physical terms, what this means is that

there is in q-deformed geometry just one self-dual

object B

(G) with two different scaling limits

UðgÞ B

ðGÞ!C½G

as q !1, and the structure of which underlies the

deeper structure of U

(g) and C

[G] as well.

Braided Groups and Quantum Planes

A super qua ntum group or super-Hopf algebra is

not a quantum group or Hopf algebra since the key

homomorphism pro perty of  : H !H  H is mod-

ified: one must use in the target H

H the Z

-graded

or super tensor product of super algebras. Here,

ða  bÞðc  dÞ¼ð1Þ

jbjjcj

ac  bd

694 Hopf Algebras and q-Deformation Quantum Groups

for elements of degree jbj, jcj. Super quantum groups

(gl

m jn

) etc., have been constructed and have an

analogous theory to the bosonic versions above.

Super spaces in physics are associated to differential

forms and in the same way a bicovariant exterior

algebra on a quantum group H is generally a super

quantum group. Here the exterior algebra is

generated on by 1-forms and the coproduct on 1-

forms is

 ¼ 

þ 

Here 

L, R

are the coactions of H on 1-forms

induced by the left and right coaction of H on itself.

For a true understanding of quantum groups one

must, however, go beyond such objects to ‘‘braided

groups’’ or Hopf algebras with braid statistics (see

Majid 1995). This theory was introduced by the

author in the early 1990s as a more systematic

method for q-deformation of structures in physics

based on q-group covariance. We have seen that a

quasitriangular quantum group, or any Hopf alge-

bra through its double, generates a braided category

with the flip map  replaced by a braiding 

V, W

between any two represent ations. Anything which is

covariant under the quantum group means by

definition that it lives in the braided category.

Working with such ‘‘braided algebras’’ is similar to

working with superalgebras except that one should

use  in place of the graded transposition in any

algebraic construction. In particular, two braided

algebras have a natural ‘‘braided tensor product’’

also in the category. In concrete terms,

ða  bÞðc  dÞ¼aðb  cÞd

Then a Hopf algebra in the braided category or

braided group is B, an algeb ra in the category along

with a coalgebra and antipode, where

 : B !BB

is an algebra homomorphism (see Braided and

Modular Tensor Categories).

Next, we have mentioned in the section

‘‘ q-Defor mation envelo ping algebras’’ that q-alge-

bras generate topological invariants, but we now

turn this on its head and use braid diagrams to do q-

algebra. We write all operations as flowing down

the page, any transpositions in the algebraic con-

struction are expressed as a braid crossing =

its inverse by the reversed braid crossing, and any

other operations as nodes. Thus, a product is

denoted

and a coproduct . Algebraic informa-

tion ‘‘flows’’ along these ‘‘wires’’ much like the way

that information flows along the wiring in a

computer, except that under- and over-crossings

represent distinct nontrivial operators. (In fact, one

may formulate topological quantum computers

exactly in this way.) In this notation, tensor

products are denoted by juxtaposition an d the trivial

object in the category is omitted. In particular, one

has the axioms and all genera l theorems of Hopf

algebras at this diagrammatic level. For example, the

adjoint action of any braided group B on itself is

(see Majid 1995)

Ad =

In any concrete example, such diagrams turn into

R-matrix formulas where =R as explained in the

section ‘‘q -Deformat ion envel oping algebras.’’

A basic example of a braided group is the braided

q-pla ne C

with generators x, y and relations

yx = qxy. Its copro duct is the additive one

x = x 

1 þ 1  x (and similarly for y) reflecting addition in

the plane, but this is extended to products as a

braided group with braiding q

1=2

1=2, 1=2

in terms of

the R-ma trix in the section ‘‘ q-Defor mation envel-

oping algebras. ’’ The extra factor here means that

lives in the braide d cate gory of representations of

(gl

) =

(sl

) (i.e., with an additional central

(1) generator to provide the q

1=2

). More precisely,

the category is that of corepresentations of

[GL

[SL

]. The coaction in this case is



ðxyÞ¼ðxyÞ



where the additional central generator is encoded in

the q-determinant (which is no longer set equal to

1). Notice that q

1=2

1=2, 1=2

has eigenvalues q, q

1

(one says that it is q-Hecke). Another braided group,

associated now to the second eigenvalue is C

0 j2

with generators ,  and relations  = q

1

,



= 

= 0. It is the quadratic algebra dual of C

(Manin 1988).

One has natural braided linear spaces for the whole

family C

[G], on which the latter coact after central

extension. The general construction is as follows. If V

is an object in a braided category (e.g., the funda-

mental representation of a quantum group), let T(V)

be the tensor algebra generated by a basis {e

}ofV

with no relations and the additive braided coproduct

as above. Assume that V has a dual V



in the

category, and similarly form T(V



) with dual basis

generators {f

}. These two braided groups will be

dually paired by ex tending the evaluation map to

products, which takes the form of ‘‘braided integers’’

(see Majid 1995)

f

; e

e

i¼

n;m

½n; !

i

j

½n; ¼id þ 

þ 



þþ



n1;n

Hopf Algebras and q-Deformation Quantum Groups 695