A quantum groups primer

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
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155 Classification theories of polarized varieties, TAKAO FUJITA
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159 Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds)
160 Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds)
161 Lectures on block theory, BURKHARD KULSHAMMER
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173 Discrete groups and geometry, W.J. HARVEY is C. MACLACHLAN (eds)
174 Lectures on mechanics, J.E. MARSDEN
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178 Lower K- and L-theory, A. RANICKI
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181 Geometric group theory I, G.A. NIBLO it M.A. ROLLER (eds)
182 Geometric group theory II, G.A. NIBLO it M.A. ROLLER (eds)

183 Shintani zeta functions, A. YUKIE
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187 Surveys in combinatorics, 1993, K. WALKER (ed)
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208 Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds)
209 Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA & N. YUI
210 Hilbert C*-modules, E.C. LANCE
211 Groups 93 Galway / St Andrews I, C.M. CAMPBELL et al (eds)
212 Groups 93 Galway / St Andrews II, C.M. CAMPBELL et al (eds)

214 Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO et al
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216 Stochastic partial differential equations, A. ETHERIDGE (ed)
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218 Surveys in combinatorics, 1995, PETER ROWLINSON (ed)
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224 Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds)
225 A mathematical introduction to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA,
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226 Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds)
227 Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI
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228 Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds)
229 Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK
230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN
231 Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds)
232 The descriptive net theory of Polish group actions, H. BECKER & AS. KECHRIS
233 Finite fields and applications, S. COHEN & H. NIEDERREITER (eds)
234 Introduction to subfactors, V. JONES & V.S. SUNDER
235 Number theory 1993-94, S. DAVID (ed)
236 The James forest, H. FETTER & B. GAMBOA DE BUEN
237 Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREAVES et al
238 Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds)
239 Clifford algebras and spinors, P. LOUNESTO
240 Stable groups, FRANK O. WAGNER
241 Surveys in combinatorics, 1997, R.A. BAILEY (ed)

242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds)
243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds)
244 Model theory of groups and automorphism groups, D. EVANS led)
245 Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al
246 p-Automorphisms of finite p-groups, E.I. KHUKHRO
247 Analytic number theory, Y. MOTOHASHI led)
248 Tame topology and o-minimal structures, LOU VAN DEN DRIES
249 The atlas of finite groups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds)
250 Characters and blocks of finite groups, G. NAVARRO
251 Grobner bases and applications, B. BUCHBERGER & F. WINKLER (eds)
252 Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO, R. STOHR (eds)
253 The q-Schur algebra, S. DONKIN
254 Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds)
255 Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds)
256 Aspects of Galois theory, HELMUT VOLKLEIN et al
257 An introduction to noncommutative differential geometry and its physical applications 2ed,
J. MADORE
258 Sets and proofs, S.B. COOPER & J. TRUSS (eds)
259 Models and computability, S.B. COOPER & J. TRUSS (eds)
260 Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al
261 Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al
263 Singularity theory, BILL BRUCE & DAVID MOND (eds)
264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds)
265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART
267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds)
268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJOSTRAND
269 Ergodic theory and topological dynamics, M.B. BEKKA & M. MAYER
270 Analysis on Lie Groups, N.T. VAROPOULOS & S. MUSTAPHA
271 Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV
272 Character theory for the odd order function, T. PETERFALVI

273 Spectral theory and geometry, E.B. DAVIES & Y. SAFAROV (eds)
274 The Mandelbrot set, theme and variations, TAN LEI (ed)
275 Computatoinal and geometric aspects of modern algebra, M. D. ATKINSON et al (eds)
276 Singularities of plane curves, E. CASAS-ALVERO
277 Descriptive set theory and dynamical systems, M. FOREMAN et al (eds)
278 Global attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO
279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds)
280 Characters and automorphism groups of compact Riemann surfaces, T. BREUER
281 Explicit birational geometry of 3-folds, ALESSIO CORTI & MILES REID (eds)
282 Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO
283 Nonlinear elasticity, R. OGDEN & Y. FU (eds)
284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SULI (eds)
285 Rational Points on Curves over Finite Fields, H. NIEDERREITER & C. XING
286 Clifford algebras and spinors 2ed, P. LOUNESTO
287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA
& E. MARTINEZ (eds)
288 Surveys in ombinatorics, 2001, J.W.P. HIRSCHFELD(ed)
289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE
290 Quantum Groups and Lie Theory, A. PRESSLEY (ed)
291 Tits Buildings and the Model Theory of Groups, K. TENT (ed)

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London Mathematical Society Lecture Note Series. 292

A Quantum Groups Primer

Shahn Majid
Queen Mary, University of London

AMBRIDGE

UNIVERSITY PRESS

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PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom
CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge C132 2RU, UK
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Dock House, The Waterfront, Cape Town 8001, South Africa


© Shahn Majid 2002

This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press
First published 2002
Typeface Computer Modern 10/13.

System LATEX 2e [Typeset by the author]

A catalogue record of this book is available from the British Library

Library of Congress Cataloguing in Publication data
ISBN 0 521 01041 1 paperback
Transferred to digital printing 2003

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For my friends

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Contents

Preface
page ix
1
Coalgebras, bialgebras and Hopf algebras. Uq(b+)
1
2
Dual pairing. SLq(2). Actions
9
3
Coactions. Quantum plane A2

17
4
Automorphism quantum groups
23
Quasitriangular structures
29
5
Roots of unity. uq(sl2)
6
34
q-Binomials
7
39
Quantum double. Dual-quasitriangular structures
44
8
Braided
categories
9
52
10
(Co)module categories. Crossed modules
58
11
q-Hecke algebras
64
12
Rigid objects. Dual representations. Quantum dimension
70
13

Knot invariants
77
14
Hopf algebras in braided categories. Coaddition on A2
84
15
Braided differentiation
91
16
Bosonisation. Inhomogeneous quantum groups
98
17
Double bosonisation. Diagrammatic construction of uq(sl2) 105
18
The braided group Uq(n+). Construction of Uq(g)
113
19
q-Serre relations
120
20
R-matrix methods
126
21
Group, algebra, Hopf algebra factorisations. Bicrossproducts 132
22
Lie bialgebras. Lie splittings. Iwasawa decomposition
139
23
Poisson geometry. Noncommutative bundles. q-Sphere
146

24
Connections. q-Monopole. Nonuniversal differentials
153
Problems
159
Bibliography
166
Index
167
vii

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Preface

Hopf algebras or `quantum groups' are natural generalisations of groups.
They have many remarkable properties and, nowadays, they come with
a wealth of examples and applications in pure mathematics and mathematical physics.
Most important are the quantum groups Uq (g) modelled on, and in

some ways more natural than, the enveloping algebras U(g) of simple Lie algebras g. They provide a natural extension of Lie theory.
There are also finite-dimensional quantum groups such as bicrossproduct
quantum groups associated to the factorisation of finite groups. Moreover, quantum groups are clearly indicative of a more general 'noncommutative geometry' in which coordinate rings are allowed to be noncommutative algebras.
This is a self-contained first introduction to quantum groups as alge-

braic objects. It should also be useful to someone primarily interested

in algebraic groups, knot theory or (on the mathematical physics side)
q-deformed physics, integrable systems, or conformal field theory. The
only prerequisites are basic algebra and linear algebra. Some exposure
to semisimple Lie algebras will also be useful.
The approach is basically that taken in my 1995 textbook, to which
the present work can be viewed as a companion `primer' for pure mathematicians. As such it should be a useful complement to that much
longer text (which was written for a wide audience including theoretical
physicists). In addition, I have included more advanced topics taken
from my review on Hopf algebras in braided categories and subsequent
research papers given in the Bibliography, notably the `braided geometry' of UQ(g). This is material which may eventually be developed in a
sequel volume to the 1995 text.
In particular, our approach differs significantly from that in other
ix

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x

Preface

textbooks on quantum groups in that we do not define Uq(g) by means
of generators and relations `pulled out of a hat' but rather we deduce
these from a more conceptual braided-categorical construction. Among
the benefits of this approach is an inductive definition of Uq(g) as given

by the repeated adjunction of `quantum planes'. The latter, as well
as the subalgebras Uq(n+), are constructed in our approach as braided
groups, which can be viewed as a modern braided-categorical setting for
the first (easy) part of Lusztig's text.

The book itself is the verbatim text of a course of 24 lectures on
Quantum Groups given in the Department of Pure Mathematics and
Mathematical Statistics at the University of Cambridge in the Spring of
1998. The course was at the Part III diploma level of the mathematics
tripos, which is approximately the level of a first year graduate course
at an American university, perhaps a bit less advanced. Accordingly,
it should be possible to base a similar course on this book, for which
purpose I have retained the original lecture numbering. The first 1/3
of the lectures cover the basic algebraic structure, the second 1/3 the
representation theory and the last 1/3 more advanced topics. There
were also three useful problem sets distributed during the course, which
I include at the end of the book.
I would like to thank the students who attended the course for their
useful comments. Particularly, the lectures start off quite slowly with

a lot of explicit computations and notations from the theory of Hopf
algebras; depending on the wishes of the students, one could skip faster
through these lectures by deferring the proofs as exercises - with solutions on handouts. Meanwhile, the last five lectures are an introduction
to some miscellaneous topics; they are self-contained and could be omitted, depending on the time available. Finally, I want to thank Pembroke
College in the University of Cambridge, where I was based at the time
and during much of the period of writing.
Shahn Majid
School of Mathematical Sciences
Queen Mary, University of London

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1

Coalgebras, bialgebras and Hopf algebras.
Uq(b+)

Quantum groups today are like groups were in the nineteenth century,
by which I mean
- a young theory, abundant examples, a rich and beautiful mathematical structure. By `young' I mean that many problems remain wide open,
for example the classification of finite-dimensional quantum groups.
- a clear need for something like this in the mathematical physics of
the day. In our case it means quantum theory, which clearly suggests
the need for some kind of `quantum geometry', of which quantum groups
would be the group objects.

These are algebra lectures, so we will not be able to say too much
about physics. Suffice it to say that the familiar `geometrical' picture
for classical mechanics: symplectic structures, Riemannian geometry,
is all thrown away when we look at quantum systems. In quantum
systems the classical variables or `coordinates' are replaced by operators

on a Hilbert space and typically generate a noncommutative algebra,
instead of a commutative coordinate ring as in the classical case. There
is a need for geometrical structures on such quantum systems parallel
to those in the classical case. This is needed if geometrical ideas such as
gravity are ever to be unified with quantum theory.
From a mathematical point of view, the motivation for quantum groups
is:

- the original (dim) origins in cohomology of groups (H. Hopf, 1947);
an older name for quantum groups is `Hopf algebras'
- q-deformed enveloping algebra quantum groups provide an expla-

nation for the theory of q-special functions, which dates back to the
1900s. They are used also in number theory. (For example, there are
q-exponentials etc., related to quantum groups as ordinary exponentials
are related to the additive group R.)
1

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1 Coalgebras, bialgebras and Hopf algebras. Uq(b+)

2

A®A®A
m®id

A®A

id®m

A®A

A

A®A

A®A

k®A =A

A®k =A

Fig. 1.1. Associativity and unit element expressed as commutative diagrams.

- representations of quantum groups form braided categories, leading
to link invariants
- quantum groups are the `group' objects in some kind of noncommutative algebraic geometry
- quantum groups are the `transformation' objects in noncommutative
algebraic geometry

- quantum groups restore an input-output symmetry to algebraic
constructions; for example, they admit Fourier theory.

We fix a field k over which we work. We begin by recalling that an
algebra A is
1. A vector space over k.

2. A map m : A ® A - A which is associative in the sense (ab)c =
a(bc) for all a, b, c E A. Here ab = m(a 0 b) is shorthand.
3. A unit element IA, which we write equivalently as a map rl : k - A

byrl(1)=1A. Werequire alA=a=lAafor allaEA.
In terms of the maps, these axioms are given by the commutative
diagrams in Figure 1.1. Note that most algebraic constructions can, like
the axioms themselves, be expressed as commuting diagrams. When all
premises, statements and proofs of a theorem are written out like this
then reversing all arrows will also yield the premises, statements and
proofs of a different theorem, called the `dual theorem'.

Definition 1.1 A coalgebra C is
1. A vector space over k.
2. A map A : C -> C ® C (the `coproduct) which is coassociative in

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1 Coalgebras, bialgebras and Hopf algebras. Uq(b+)

3

C®C®C

id®0

Aoid

C®C

C®C

C®C

C®C

k®C =C

C

Fig. 1.2. Coassociativity and counit element expressed as commutative diagrams.

the sense

E C(1) (1)

® C(1) (2) ® C(2) _

E C(1) ®

C(2) (1) ® C(2) (2)

for all c E C. Here Ac j c(1) 0 c(2) is shorthand.
3. A map e : C -+ k (the `count') obeying > e(c(1))c(2) = C =
C(1)E(C(2)) for all c E C.

In terms of the maps, these axioms are given by the commutative
diagrams in Figure 1.2, which is just Figure 1.1 with all arrows reversed.

This notion of reversing arrows has the same status as the idea, familiar in algebra, of having both left and right module versions of a
construction. The theory with only left modules is equivalent to the
theory with right modules, by a left-right reflection (i.e. reversal of tensor product). But one can also consider theorems with both left and
right modules interacting in some way, e.g. bimodules. Similarly, the
arrow-reversal operation transforms theorems about algebras to theorems about coalgebras. However, we can also consider theorems involving both concepts. In this way, quantum group theory is a very natural
`completion' of algebra to a setting which is invariant under the arrowreversal operation.

Definition 1.2 A bialgebra H is
1. An algebra H, m, g.
2. A coalgebra H, A, E.
3. 0, e are algebra maps, where H 0 H has the tensor product algebra

structure (h 0 g) (h' ®g') = WO gg' for all h, h', g, g' E H.

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1 Coalgebras, bialgebras and Hopf algebras. U9(b+)

4

H®Hm- H
A®0
I

H®H®H®H

HEk

'' H®H
m®m

is®T®sa

I

_ H®H®H®H

H®H

k

77

H

1®E

IM

H®H '

id®S S®id

\,7 (g,7 1,N
`H®H

H®H

Fig. 1.3. Additional axioms that make the algebra and coalgebra H into a
Hopf algebra.

Actually, a bialgebra is more like a quantum `semigroup'. We need
something playing the role of group inversion:

Definition 1.3 A Hopf algebra H is
1. A bialgebra H, A, e, m, g.

2. A map S : H --+ H (the `antipode') such that E(Sh(,))h(2) _
e(h) = E h(,)Sh(2) for all h E H.

The axioms that make a simultaneous algebra and coalgebra into a
Hopf algebra are shown in Figure 1.3, where T : H ® H - H ® H is the

`flip' map T(h ®g) = g ®h for all h, g E H.
Proposition 1.4 (Antihomomorphism property of antipodes). The antipode of a Hopf algebra is unique and obeys S(hg) = S(g)S(h), S(1) = 1
S is an antialgebra map) and (S ®S) o Ah = T o A o Sh, Sh = eh
(i.e. S is an anticoalgebra map), for all h,g E H.
(i.e.

Proof During proofs, we will usually omit the E signs, which should be
understood. If S, Sl are two antipodes on a bialgebra H then they are
equal because Sih = (Sih(,))e(h(2)) = (Sih(l))h(2) (1) Sh(2)(2) = (Sih(l)(l))
h(j) (2)Sh(2) = e(h(l))Sh(2) = Sh. Here we wrote h = h(j)6(h(2)) by the
counit axioms, and then inserted h(2) (,) Sh(2) (2) knowing that it would
collapse to e(h(2)). We then used associativity and (the more novel ingredient) coassociativity to be able to collapse (Slh(l)(l))h(,) (2) to e(h(,)).
Note that the proof is not any harder than the usual one for uniqueness

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1 Coalgebras, bialgebras and Hopf algebras. Uq(b+)

5

of group inverses, the only complication being that we are working now
with parts of linear combinations and have to take care to keep the or-

der of the coproducts. We can similarly collapse such expressions as
(S1h(1))h(2) or h(2)Sh(3) wherever they occur as long as the two collaps-

ing factors are in linear order. This is just the analogue of cancelling
h-1h or hh-1 in a group. Armed with such techniques, we return now
to the proof of the proposition. Consider the identity

(S(h(1)(1)9(1)(1)))h(1)(2)9(1)(2) ®9(2)0 h(2)

_ (S((h(1)9(I))(1)))(h(I)9(1))(2) ®9(2) ®h(2)

= 1®g ®h.

= e(h(1)9(1))1®9(2)0 h(2)

We used that A is an algebra homomorphism, then the antipode axiom
applied to h(1)g(1). Then we used the counity axiom. Now apply S to
the middle factor of both sides and multiply the first two factors. One
has the identity
Sg®h = (^'(."(1)(1)9(1) (1)))h(1) (2)9(1) (2) S9(2) ® h(2)
= (S(h(1)(1)9(1)))h(1)(2)9(2)(1)S9(2)(2)

®h(2) =

(S(h(1)(1)9))h(1)(2) ®h(2),

where we used coassociativity applied to g. We then use the antipode
axiom applied to 9(2), and the counity axiom. We now apply S to the
second factor and multiply up, to give
(Sg)(Sh) = (S(h(1)(1)9))h(1)(2)Sh(2) = (S(h(1)9))h(2)(1)Sh(2)(2)

= S(hg).

We used coassociativity applied to h, followed by the antipode axioms
applied to h(2) and the counity axiom.

Example 1.5 The Hopf algebra H = Uq(b+) is generated by 1 and the

elements X, g, g-1 with relations
99-1 = 1 = 9-19,

9X = qXg,

where q is a fixed invertible element of the field k. Here

AX =X®1+g®X, Ag=g®9, 09-1 =9-1®g-1,
eX = 0,

eg = 1 =

Note that S2

eg-1

SX = _g-1X

Sg = g-1

Sg-1

= g.

id in this example (because S2X = q-1X).

Proof We have A, e on the generators and extend them multiplicatively to products of the generators (so that they are necessarily algebra

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1 Coalgebras, bialgebras and Hopf algebras. Uq(b+)

6

maps as required). However, we have to check that this is consistent
with the relations in the algebra. For example, LgX = (Og)(OX) =
(g (D g) (X (& 1 + g ®X) = gX ®g + g2 ®gX , while equal to this must be

OqX g = q(OX) (Og) = q(X ®1 + g ®X) (g ®g) = qX g ®g + qg2 ®Xg.
These expressions are equal, using again the relations in the algebra as

stated. Similarly for the other relations. For the antipode, we keep in
mind the preceding proposition and extend S as an antialgebra map,
and check that this is consistent in the same way. Since S obeys the
antipode axioms on the generators (an easy computation), it follows
that it obeys them also on the products since A, e are already extended
multiplicatively.

It is a nice exercise - we will prove it later in the course, but some
readers may want to have fun doing it now - to show that

AX, =

m
[

Xrn'-rgr ®Xr

]

r

r=0

q

where
[m]gi

m
I

r

Iq

[r]q! [m - r]q!

'

[r]q! = [r]q[r - 1]q ... [1]q

are the q-binomial coefficients defined in terms of `q-integers'
[r]q=1+q+...+qr-1=

1-qr

1-q

The last expression here should be used only when q # 1, of course. We
should also assume [r]q are invertible to write the q-binomial coefficients
in this way.

Example 1.6 Let G be a finite group. The group Hopf algebra kG is
the vector space with basis G, and the algebra structure, unit, coproduct,
counit and antipode

product in G,

1 = e,

Og = g ®g,

eg = 1,

Sg = g-1

on the basis elements g E G (extended by linearity to all of kG).

Proof The multiplication is clearly associative because the group multiplication is. The coproduct is coassociative because it is so on each
of the basis elements g E G. It is an algebra homomorphism because

0(gh) = gh ®gh = (g ®g) (h ®h) = (Og) (Oh). The other facts are
equally easy. G does not actually need to be finite for this construction,
but we will be interested in the finite case.

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1 Coalgebras, bialgebras and Hopf algebras. Uq(b+)

7

So all of finite group theory should, in principle, be a special case
of Hopf algebra theory. The same is true for Lie theory, if we use the
enveloping algebra. We recall that a Lie algebra is:
1. A vector space g.
2. A map [ , ] : g 0 g -> g obeying the Jacobi identity and antisymmetry axioms (when the characteristic of k is not 2).

Example 1.7 Let g be a finite-dimensional Lie algebra over k. The
universal enveloping Hopf algebra U(g) is the noncommutative algebra
generated by 1 and elements of a basis of g modulo the relations [i;, 77] =
t 77 - q for all l;, r7 in the basis. The coproduct, counit and antipode are

0=t;®1+1®t;,

et;=O,

St;=-1=

extended in the case of A, c as algebra maps, and in the case of S as an
antialgebra map.
Proof We extend A, e as algebra homomorphisms and S as an antialgebra homomorphism, and have to check that this extension is consistent
with the relations. For example, A(1 77) = ( (9 1 + 1®t;) (r7 ®1 + 1® 77) =
r7
®Z; Subtracting from this the corresponding expression for Or7 and using the relations, we obtain [i;, i7] ®1 + 1®[Z;, 77] _
A[l;, 77] as required. Similarly for the counit and antipode.

One can say, informally, that U(g) is generated by 1 and elements of

g with the relations stated; it does not depend on a choice of basis. A
more formal way to say this is to construct first the tensor Hopf algebra
T (V) = k ® V ® V ® V ® V ® V 0 V ®. . . on any vector space V. The
product here is (v(D ...®w)(x(D
(v(D .
This forms a Hopf algebra with

Ov=v®1+10v, ev=0, Sv=-v
for all v E V. The enveloping algebra U(g) is the quotient of T(g)
modulo the ideal generated by the relations 0,q - ®t; = [t;,
(Of
course, the best definition is as a universal object, but we will not need
that.)

So Lie theory is also contained, in principle, as a special case of
quantum group theory. In fact, one of the main motivations for Hopf
algebras in the 1960s was precisely as a tool that unifies the treatment
of results for groups and Lie algebras into one technology, e.g. their cohomology theory. Clearly, our example Uq(b+) is a mixture of these two

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8

1 Coalgebras, bialgebras and Hopf algebras. Uq(b+)

kinds of `classical' Hopf algebras. It has an element g which is grouplike

in the sense that it obeys Ag = go g. And it has an element X which
is a bit like the Lie case. But it is neither a group algebra nor an enveloping algebra exactly. What characterises these classical objects, in

contrast to Uq(b+), is:

Definition 1.8 A Hopf algebra is commutative if it is commutative as
an algebra. It is `cocommutative' if it is cocommutative as a coalgebra,
i.e. if r o A = A. This is the arrows-reversed version of commutativity.
Corollary 1.9 If H is a commutative or cocommutative Hopf algebra,
then S2 = id.

Proof We use Proposition 1.4, so that S2h = (S2h(,))(Sh(2))h(3) =
(S(h(1)Sh(2))) h(3) = h in the cocommutative case. Here we use a neutral
notation h(l) ® h(2) ® h(3) = h(1)(1) ® h(1)(2) ® h(2) = h(l) ® h(2)(1) ® h(2)(2)

(just as one writes abc - (ab)c = a(bc)). The other case is similar.

Clearly, kG and U(g) are cocommutative. The coordinate rings of
linear algebraic groups are likewise commutative Hopf algebras, while

Uq(b+) is neither. As a tentative definition, we can say that a truly
`quantum' group (in contrast to a classical group or Lie object viewed
as one) is a noncommutative and noncocommutative Hopf algebra. Later
on, we will add further properties as well.

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2

Dual pairing. SLq (2) Actions
.

In the last lecture we showed how to view finite groups and Lie algebras

as Hopf algebras, and gave a variant that was truly `quantum'. We
now complete our basic collection of examples with some other classical
objects.

Example 2.1 Let G be a finite group with identity e. The group function
Hopf algebra k(G) is the algebra of functions on G with values in k and
the pointwise product (fg)(x) = f (x)g(x) for all x E G and f, g E k(G).
The coproduct, counit and antipode are
(Of) (x, y) = f (xy),

e f = f (e),

(Sf) (x) = f (x-1),

where we identify k(G) ® k(G) = k(G x G) (functions of two group variables).

Proof Coassociativity is evidently ((A ®id)A f) (x, y, z) = (A f) (xy, z) _

f ((xy)z) = f (x(yz)) = (Af)(x, yz) = ((id 0 A)Af)(x, y, z). Note that
it comes directly from associativity in the group. Likewise, the counity
and antipode axioms come directly from the group axioms for the unit
element and inverse.
Also, when g is a finite-dimensional complex semisimple Lie algebra
(as classified by Dynkin diagrams), it has an associated complex Lie

group G C .,,,C) (the n x n matrices with values in C). This subset is
of the form G = {x E MM I p(x) = 0}, where p is a collection of polynomial equations. Correspondingly, we have an algebraic variety with
coordinate algebra C[G] defined as C[x2j] where i,j = 1, ... , n (polynomials in n2 variables), modulo the ideal generated by the relations

p(x) = 0. The group structure inherited from matrix multiplication
9

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2 Dual pairing. SLq(2). Actions

10

corresponds to a coproduct and counit
Ox2.j = E xzk ®xk,j ,

e(x23) = b2J

k

where Sj is the Kronecker delta-function. There is also an antipode
given algebraically via a matrix of cofactors of the matrix xzj of generators. In this way, we have a complex linear algebraic group with
coordinate algebra (C[G] as a Hopf algebra. In fact, G can be taken so
that the coefficients of p(x) are integers (from work of Chevalley) giving a
coordinate ring Z[G]. Then, by tensoring with k, the same construction
works over any field and provides a Hopf algebra k[G] (and considering
all k, one has an affine group scheme).

Example 2.2 The Hopf algebra k[SL2] is k[a, b, c, d] modulo the relation
a

b

c

d

det

= 1.

The coproduct, counit and antipode are

Aa=a®a+b®c, Ob=b®d+a®b, Oc=c®a+d®c,
E(a)=E(d)=1,

Od = d ®d + c ®b,

E(b)=E(c)=O,

Sa=d, Sd=a, Sb=c, Sc=b.
The coalgebra and antipode here can be written more concisely as

0

a

b

(a

(a

b

b

®
c

d

c

d

,

c

a

b

c

d

d
d

-b

-c

a

=

S

a

b

1

0

c

d

0

1

e

,

where matrix multiplication should be understood in this definition of
0. This is no more than a shorthand notation. Finally, for a truly

`quantum' variant of this:

Example 2.3 Let q E k*. The Hopf algebra SLq(2) is k(a, b, c, d)
(the free associative algebra) modulo the ideal generated by the six `q-

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2 Dual pairing. SLq(2). Actions

11

commutativity' relations

ca = qac,

ba = qab,

do = qcd,

db = qbd,

be = cb,

da-ad=(q-q-1)bc
and the `q-determinant' relation

ad - q-'bc= 1.
The coalgebra has the same matrix form on the generators as above, and
the antipode is

Sd = a,

Sb = -qb,

So = d,

Sc = -q-1c.

Proof We will give general constructions for this kind of quantum group

later on. For the moment, it is easy enough to verify directly that it
fulfils the axioms. Hint: first consider the algebra Mq(2) defined in the
same way but without the q-determinant relation. This is a quadratic
algebra and it is easier to verify that 0, e are well-defined when extended

to products. Then show that ad - q-1bc is central in this algebra and
grouplike in the sense 0(ad - q-1bc) = (ad - q-1bc) ®(ad - q-1bc) and
e(ad - q-1bc) = 1. The further relation ad - q-1bc = 1 can then be
added and the quotient remains a bialgebra. For the antipode, it is easy
enough to see that it extends antimultiplicatively. Once well-defined,
it is enough to check the antipode axioms on the generators, which is
elementary. Note that in the compact `matrix' notation one writes

ad - q-1bc =_ det
4

a

b

,

c

d

S

a

b

c

d

=

d

-q b

-q-1c

a
O

The quantum group SLq (2) here is also variously denoted kq [SL21 or

°q(SL2) in the literature. It completes our collection of basic examples. Here kG, k(G) for finite groups and U(g), k[G] for Lie algebras are

`classical' objects, while Uq(b+) and SLq(2) are more novel and truly
`quantum' groups according to the tentative definition given at the end
of the last lecture. We now return to the general theory of Hopf algebras.

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12

2 Dual pairing. SLq(2). Actions

Definition 2.4 Two Hopf algebras H, H' are `dually paired' by a map

(, ):H'®H-*kif

(1,h) = E(h)

(o4, h ®g) _

hg),

(SO, h) = (0, Sh)

for all 0, 0 E H' and h, g E H. Here (, ) extends to tensor products
pairwise.

This says that the product of H and coproduct of H' are adjoint to
each other under ( , ), and vice-versa. Likewise, the units and counits
are mutually adjoint, and the antipodes are adjoint. The definition is
made possible by the invariance of the Hopf algebra axioms under arrow-

reversal (i.e. input-output symmetry) as explained in the last lecture.
If H is finite dimensional, then ( , ) = ev (the evaluation map) provides a duality pairing with H*. Here, H* has the product A* and the
coproduct m*, where
A* : (H ®H)* -* H*,

m* : H - (H (9 H)*

are the duals of A, ,m of H. They define the required maps since
(H ® H)* D H* ® H* is an equality for a finite-dimensional vector space

H (otherwise, it need not be an equality and m* need not descend to
a coproduct on H*). This is the unique possibility for a nondegenerate
duality pairing in the finite-dimensional case, and we say H* is the dual
Hopf algebra in this case.
Among our examples, k(G)* = kG (by evaluation) and U(g), C[G] are
dually paired over C for g a finite-dimensional complex semisimple Lie
algebra. If p : g C M,, (C) is the defining representation of g, the pairing
is

x'j) = p(e)'j,

VE g.

This result also extends to a general field with both g and p defined over
k. Meanwhile, Uq(b+) is self-dual:

Proposition 2.5 Uq(b+) is dually paired with itself by
(g, g) = q,

(X, X) = 1,

(X, g) = (g, X) = 0.

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2 Dual pairing. SLq(2). Actions

13

Proof We will see general methods for this kind of result later in the
course. For the moment, it is a nice exercise directly from the definitions.
Hint: first find that f,,,,,n(9) - (Xmgn,g) = (Xm,g)(gn,g) = gnd,,,,,o and

fm,n(X) - (Xmgn,X) _ (Xm,X)(gn,1) = lsm,1. Then the coproduct
A(Xmgn) = ( Xm)(gn ®gn) given in the last lecture, and the axioms
of a pairing, imply that
m

f ll fm-r,n+r(h)fr,n(h')
fm,n(hh') _
Lm] q
r=0
for all h, h' E Uq(b+). This determines fm,n on products, which shows
that ( , ) is uniquely determined. We then define it on the basis
{XmgnI n E Z, m E Z+} of each copy of Uq(b+) (where Z+ includes 0),
by the resulting formula for fmn, and verify the duality pairing axioms
on products and coproducts of basis elements.

Finally, by definition, an action or representation of a bialgebra or
Hopf algebra H means one of the underlying algebra. What is special
about having a bialgebra is that one may tensor product representations.
Clearly, if V, W are H-modules (i.e. H acts on them), then
hi (v ®w) = E h(,)> v ®h(2)Nw - (Oh)>(v ®w)

for all h E H and v c V, w E W, makes V ®W into a H-module. Here
is used to denote a left action. One always has a trivial module V = k,
with
WA = e(h)A, Vh E H, A E k.

This is the identity object under the tensor product of modules.

Definition 2.6 A bialgebra or Hopf algebra H acts on an algebra A
(one says that A is an H-module algebra) if
1. H acts on A as a vector space.
2. The product map m : A& A -> A commutes with the action of H.
3. The unit map q : k -4 A commutes with the action of H.
Explicitly, the conditions 2,3 are

hi(ab) = E(h(,)I'a)(h(2).b),

hi'l = e(h)1,

ba, b E A,

h E H.

We leave it as an easy exercise to see what these conditions mean for
our basic examples. One finds, for all a, b E A:

(i) for kG,
g'(ab) = (gr'a)(gr.b),

g>1 = 1,

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Vg E G,


2 Dual pairing. SLq(2). Actions

14

which is the usual notion of a group action by automorphisms.
(ii) for U(g),
$$1 = 0,

l;r>(ab) _

VV E g

which is the usual notion of a Lie action by derivations.
(iii) for Uq(b+),
gt>(ab) _ (g>a)(gcb),

X'(ab) _ (Xca)b+ (g.a)(Xrb),

g.1=1, X'1=0.
One says that X acts as a 'skew-derivation'.

(iv) for k(G), it means A is a G-graded algebra, where fi(a) = f (JaI)a
on homogeneous elements of degree jal. Here an action of k(G) on a
vector space V is the same thing as a G-grading V = ®9EG V9, where
we say that I v I = g for all vE V9.
The situation for k[SL2] is roughly similar to (iv), but is not usually
considered in any context that I know of; likewise for SLq(2).

Proposition 2.7 (Adjoint action). Every Hopf algebra H acts on itself
as an algebra by

Adh(g) = E h(,)gSh(2)
for all h, g E H.

Proof We check hr(gia) = h>(g(,)aSg(2)) = h(,)g(,)a(Sg(2)) (Sh(2)) _
(hg)(,)aS(hg)(2) _ (hg)>a using Proposition 1.4 about the antipode.

Also, Ira = laS(1) = a. Thus, we have an action. We have a module algebra because h'(ab) = h(,)ab(Sh(2)) = h(,)a(Sh(2))h(3)bSh(,) =
(h(,)'a)(h(2).b) and h>1 = h(,)lSh(2) = e(h). We insert (Sh(2))h(3),
knowing that it collapses using the antipode axioms, and freely renumber to express coassociativity. Here h(l) ® h(2)®h(3)0 h(4 is our neutral
notation denoting any of the five expressions (A ® id ® id) (A ® id)Oh,
(id ®0 ® id) (A (9 id)Oh, etc. coinciding through coassociativity.

For our standard examples, we have (immediately from the definitions):

(i) for kG,

Ad9(h) = ghg-1,

Vg, h E G.

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2 Dual pairing. SLq(2). Actions

15

(ii) for U(g),

Adg(h) = h -

VVEg, hEU(g).

(iii) for Uq(b+),
Adg(h) = ql hl h,

Adx(h) = Xh - qI hI hX

for all h E Uq (b+) of homogeneous degree I h I in X.

(iv) for k(G) and k[G], the adjoint action is trivial because these
algebras are commutative, so that Ad collapses by the antipode axioms.
(v) for SLq(2), one finds for example

Adb(am) = (1 -

q)q-mbar''+i.

This action has no classical meaning in geometry or algebraic geometry,
because it would be trivial when q = 1 (the commutative case); it is our

first example of a `purely quantum phenomenon'. Nevertheless, if we
work over C for example, the action of the rescaled generator b/(q - 1)
as q -* 1 leaves a nonzero classical `remnant' which can still be useful.
[For example, the action of the special conformal transformations on

classical R' can be similarly be expressed as the remnant as q -+ 1 of
the adjoint action of a suitable q-deformed R4 on itself.]

Proposition 2.8 (Left coregular action) If H' is dually paired with a
bialgebra or Hopf algebra H, it acts on it by
R0* (h) _

(0, h«))

for all 0E H', h E H.
Proof It is easy to see that we have an action. It respects the product beg)
cause *(hg) = (hg)a)(O, (hg)ô)) = h(j)g(l)(0, h(2)g(2)) =

and 4>1=1(Â,1)=e(O),for all 0EH'and h,gEm H.
For our standard examples, we have (immediately from the definitions):

(i) for k(G) acting on kG,

R ,(g) = 0(g)g,

Vg E G, 0 E k(G).

(ii) for C[G] acting on U(g),
RX*T, (

) = 1P( )Zj + Vj, d E g.

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