A quantum groups primer

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J Hitchin, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom The titles below are available from booksellers, or, in case of difficulty, from Cambridge University Press 46 p-adic Analysis: a short course on recent work, N KOBLITZ 59 Applicable differential geometry, M CRAMPIN & F.A.E PIRANI 66 Several complex variables and complex manifolds II, M.J FIELD 86 Topological topics, I.M JAMES (ed) 88 FPF ring theory, C FAITH & S PAGE 90 Polytopes and symmetry, S.A ROBERTSON 96 Diophantine equations over function fields, R.C MASON 97 Varieties of constructive mathematics, D.S BRIDGES it F RICHMAN 99 Methods of differential geometry in algebraic topology, M KAROUBI & C LERUSTE 100 Stopping time techniques for analysts and probabilists, L EGGHE 104 Elliptic structures on 3-manifolds, C.B THOMAS 105 A local spectral theory for closed operators, I ERDELYI & WANG SHENGWANG 107 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(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) www.pdfgrip.com London Mathematical Society Lecture Note Series 292 A Quantum Groups Primer Shahn Majid Queen Mary, University of London AMBRIDGE UNIVERSITY PRESS www.pdfgrip.com 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 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.carribridge.org © 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 521 01041 paperback Transferred to digital printing 2003 www.pdfgrip.com For my friends www.pdfgrip.com www.pdfgrip.com Contents Preface page ix Coalgebras, bialgebras and Hopf algebras Uq(b+) Dual pairing SLq(2) Actions Coactions Quantum plane A2 17 Automorphism quantum groups 23 Quasitriangular structures 29 Roots of unity uq(sl2) 34 q-Binomials 39 Quantum double Dual-quasitriangular structures 44 Braided categories 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 www.pdfgrip.com www.pdfgrip.com 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 www.pdfgrip.com x Preface textbooks on quantum groups in that we 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 www.pdfgrip.com 24 Connections q-Monopole Nonuniversal differentials 155 For classical homogeneous spaces the existence of a canonical connect- ion is based on local triviality; one builds it near the identity and then translates it around to other coordinate patches In our case, we obtain the corresponding result by entirely algebraic methods, which is a new proof even for q = 1: Example 24.4 The quantum Hopf fibration SLQ(2) - C[g,g-1] with base S9 is a quantum principal bundle, and has canonical connection (the q-monopole) defined by i(gn - 1) = a', 2(g-n - 1) = dn Explicitly, w(g - 1) = dda - qbdc Proof (Sketch) we verify the conditions in the lemma It is then elementary to compute w El When q = we recover the canonical U(1) connection (the charge monopole) over the sphere This is a nontrivial connection with nontrivial Chern class etc Finally, all of the above was with the universal differential calculus Such calculi are used in algebraic topology but are much bigger than their classical counterparts even when the algebra is commutative We now conclude with an introduction to the more interesting nonuniversal differential calculi Actually, even for an ordinary manifold there is not a unique differential structure (although one tends to use standard ones) The nonuniqueness is even more pronounced in the quantum case where we have weaker axioms (different left and right actions on the differentials) In the classical case of a Lie group the situation is better and one has a unique translation-invariant differential calculus We likewise have a better situation in the quantum group case, although usually without uniqueness Definition 24.5 A differential calculus 121 on a Hopf algebra H is bicovariant if Ill is a bicomodule, by left and right coactions 13L : SZ1 -> H ®1 l' and ,3R : 111 - 121®H /3L, OR are bimodule maps, where H ®121 and 121 ® H have the tensor product action and H acts on itself by multiplication www.pdfgrip.com 156 2.4 Connections q-Monopole Nonuniversal differentials d is a bicomodule map, where H coacts on itself by A A bicomodule is just like a bimodule with arrows reversed (the coactions commute) Also note that the above definition of a bicovariant calculus is completely left-right symmetric However, for the following classification theorem we have to make a choice to emphasise left or right comodules To fit with conventions earlier in the course we chose left (in the theory of principal bundles one might prefer the right comodule setting, for example) Lemma 24.6 Let H be a Hopf algebra Then 1l'H=kere®H by h®g h(l) ®h(2 g for all h®g E f21H ker e E HM, the braided category of crossed modules, by the left regular action and AdL(h) = E h(,) Shp) ®h(2) Proof This is elementary For the first part, we have a similar isomorphism H ® H=H ® H, which we restrict to 121 H = ker m The inverse is h ®g H h(l) ®Sh(2)g For the second part, we already know from Corollary 10.5 that H E H.M by multiplication and Ad, and we restrict this to kere Theorem 24.7 Bicovariant S21 are in 1-1 correspondence with quotient objects Al of kere E HM (i.e with AdL-stable left ideals contained in kere.) Proof (Sketch) A quotient object A' is by definition of the form A' = ker e/.M for some M which is a left ideal (to be stable under the left action) and stable under AdL in the sense AdL (M) C H ® M Given such an A1, we define Sh = Al®H, dh = (ir id) (Ah - 1®h), bh E H, where 7r : ker e - A' is the canonical projection The right (co)module structures on Sl' are those of H alone by right (co)multiplication The left (co)module structures are the tensor product of those on A' as an object of HM (as inherited from ker c in the above lemma) and those on H by left (co)multiplication The map d shown is dh = h ®1-1®h E S21H mapped under the isomorphism in the lemma and projected to Al H (This is actually -d in our previous conventions above.) In the converse direction, we know that 121 = 121H/N for some N and show that for a bicovariant calculus, M has the form M under the isomorphism in www.pdfgrip.com 24 Connections q-Monopole Nonuniversal differentials 157 the lemma, for some AdL-stable left ideal M Geometrically, one should understand A' as the space of right-invariant 1-forms associated to any calculus Q' Before giving a couple of concrete examples of a complete classification, let us note that there is an obvious notion of morphisms between bicovariant differential calculi - maps commuting with the (co)module structures and forming a commutative triangle with d One says that a calculus is irreducible (or `coirreducible' would be a more precise term) if it has no proper quotients Example 24.8 For H = k[x] with Ox = x ®1 + 10 x, the irreducible bicovariant S21 are in 1-1 correspondence with irreducible monic polyno- mials m E k[x], and take the form Q' = ka[x] where ka = k[A]/(m) is the corresponding separable field extension The bimodule structures and d are (f q5) (A, x) = f (x + A)O(A, x), (0f) (A, x) = O(A, x)f (x), d f = f(x+a)-f(x) for all f e k[x], ¢ E ka[x] Proof Here AdL is trivial so, by the theorem, bicovariant differential calculi on k[x] are in 1-1 correspondence with ideals M c ker e = (x) (the ideal generated by x in k[x]) Since k[x] is a PID, the ideal M is generated by a polynomial Since M C ker e, this polynomial is divisible by x, i.e M = (xm) for some in In this way, irreducible calculi correspond to m irreducible and monic We identify the corresponding A' = (x)/(xm) with ka = k[\] by A'ka, xf (x) - f (A) Under this identification, Q1 = A' k[x]=ka[x] The action from the right is by the inclusion k[x] C kA[x] The action from the left is by f(x) xm®xn = f(x®1+1(9 x)xm®xn as the tensor product action Hence f f (.x+x) m-lxn under our identification These expressions are modulo (xm(x)) or (m(A)) Finally, we compute d f = f (x (9 + 1®x) - 1® f (x) modulo (xm) in the first tensor factor Under our isomorphism this is the expression www.pdfgrip.com 158 24 Connections q-Monopole Nonuniversal differentials stated, modulo (m(a)) Note that dx = x modulo (xm) corresponds to E k, \[x] under our identification For a concrete example, the bicovariant calculi on C[x] are parameterised by Ao E C (say) Here m(A) = A - Ao and 7r(A) = A0 Hence, df = dx f(x + A0) - f(x) Ao (understood as dx times the coefficient of A in f (x+A) followed by setting A = A0) This is a complete classification and only the case A0 = is the standard translation-invariant calculus discovered by Isaac Newton; for Ao # one has f dg (dg) f for polynomials f, g (a bimodule structure with different left and right actions), but it is a perfectly good quantum differential calculus and actually much better behaved Example 24.9 For H = k(G) on a finite group G, the irreducible SI' are in 1-1 correspondence with nontrivial conjugacy classes C C G The 1-forms ea = >gcG(dSag)Sg for a E C form a basis of Al and f'ea = eaLa(f), df = E ea(La(f) - f)' aEC Proof This is immediate from the theorem above For A' we set to zero all delta-functions except {8a}aEC These project to our basis {ea}, which we identify in terms of d Then S21 is a free right k(G)-module with basis {ea} and we give the left module and d in these terms Here La* (f) = f (a( ))' To round off the course we note that the dual of Al is typically a braided-Lie algebra as in Figure 14.5 In the above example it has basis {xa}aEC with T (xa ® Xb) = Xb ® Xa, [Xa, Xbl = xaba-1) Lxa = Xa xa, Exa = Meanwhile for SLq(2) the S21 essentially correspond to irreducible representations of U,(812) The 2-dimensional one leads to a 4-dimensional A' Its dual is the braided-Lie algebra s12,q www.pdfgrip.com Problems Problems I * questions are optional If a Hopf algebra is commutative, show that S2 = id 2.* In any Hopf algebra, show that AoS = 7-o(S®S)o0 and coS = c, where T is the twist map (i.e an anticoalgebra map) In any Hopf algebra, show that AdR(h) = hc2> ®(Sh(j))h(3) makes H into a right H-comodule coalgebra You may assume the result of Q2 If V, W are right comodules, show that their tensor product is also a comodule in a natural way Show that if H acts from the left on a vector space V then it also acts on V*, by (hr'f)(v) = f((Sh)r'v) for all h E H, v E V and f E V* Show that a E H is central ifAdh(a) = c(h)a for all h E H Compute the left action Adh(g) = E h(1)gSh(2) for (i) kG (G a finite group) (ii) U(g) (give the action of the Lie algebra g) (iii) Uq(b+) 8.* Let [ , ] : H ®H -* H be the bilinear map [h, g] - Adh(g) (a convenient notation) Show that [x, [y, z]] = E[[xcl>, y], [x(2), z]], `dx, y, z E H (the `pentagonal Jacobi identity') What does it reduce to on the generators when H = U(g)? Let G be a finite group Show that an action of k(G) on a vector space is the same thing as a G-grading What does a module algebra under k(G) mean? (Hint: consider the action of the Kronecker functions S9 for g E G.) 159 www.pdfgrip.com Problems 160 10 Compute the left coregular action R* (h) _ h(l) (0, h(2)) for (i) k(G) acting on kG (ii) kG acting on k(G) (iii) U(g) acting on C[G] (iv) Uq(b+) paired with itself (in (iii)-(iv) the action of generators is enough) (Hint for (iv): use the module algebra property.) 11 Let q E V On polynomials k[x], show that the operation aq(xn) _ [n]gxn-1 is a q-derivation in the sense aq(fg) _ (agf)g + f(gx)agg, df,g E k[x] Hence or otherwise, show that k[x] is a Uq-i (b+)-module algebra where X acts by aq Check that the same holds on formal powerseries kQxj Assuming that [n]q # for all n E N (one says that q is `generic'), observe that age9x = xegx for all a E k Hence or otherwise, show that e9 has inverse e_X q-1- 12.* Obtain a description of the self-duality pairing (Xmgn, X,gs) of Uq(b+) with itself in terms of aq acting on k[X] 13 Show that Mq(2) defined as k(a, b, c, d) modulo the relations ca = qac, ba = qab, be=cb, db = qbd, dc = qcd, da - ad= (q-q-')bc a b c d is a bialgebra with the matrix coalgebra on the generators Show that the element ad - q-lbc is central and grouplike in Mq(2) 14.* Let H be a finite-dimensional Hopf algebra and view R E H ® H as a map H* -p H sending t- + (0 id) (R) Cast the quasitriangularity axioms (A (& id)R = R13R23 and (id ® O)R = R13R12 in terms of this map 15 Show that Cg2Z/n is factorisable for q a primitive n'th root of unity and n > odd (Hint: when m runs through Z/, so does 2m because n is odd.) 16.* If H is a bialgebra and x E H ® H is invertible and obeys X12(A®id)x = X23(id ®O)X and (e ®id)X = 1, show that HX, defined as the same algebra as H but with the new coproduct OX(h) = x(Ah)x-1 for all h E H, makes HX a bialgebra This is called the 'twisting operation' among Hopf algebras What is the equation for x when H = k(G)? (Answer: a group 2-cocycle.) 17 Let q E k* Define the q-binomial coefficients inductively by www.pdfgrip.com Problems In] 161 In [n m-1 q + InM ]q and [0]q = (and [M ]q = when m > n) Show that [m]q[n - m]q []q = [n]q (Hint: show first that In,Jq = q n-111 [ n-1 [m]q [ n],=M-1 q 18 Show that eq +B = e9 e9 where A, B obey BA = qAB and are jointly nilpotent in the sense that there exists N such that AmBn = for all m, n such that m + n = N Assume that [m]q # for all m < N 19 In Uq(sl2), show that the element qg-1 + q-1g + (q - q-1)2EF is central (this is called the q-Casimir) 20.* Let q be a primitive odd n'th root of unity Show that {gaEbFc a, b, c = 0, , n - 1} is a basis of uq(sl2) 21.* For H' dually paired with H, the quantum double H'°PmH is built on the vector space H'0 H with the product (0 ® h)(b ®g) = E' (2)o ® h(2)g(Sh(,), 3(1)) (h(3), b(3)) for all 0, b E H', and h, g E H Show that this is associative www.pdfgrip.com Problems 162 Problems II * questions are optional Show that R E Mn M,,, obeys R12R13R23 = R23R13R12 (the Yang-Baxter equations) if 41 = T o R E MM,, A,, obeys the braid rela- tions WI'124'231I'12 = `I'23W12'I'23 Here the suffixes refer to the position in Mn ® Mn, ® M,,, and T is the permutation operator k' (8) V -* kn ® kn viewed in Mn ® Mn (Hint: T12R23T12 = R13 etc.) If H, R is quasitriangular, check that 41 as in Proposition 10.1 obeys 'I'v ® w,z = `I'v,z oWw,z (completing the proof that H.M is braided) 3.* Let H, R be a dual quasitriangular Hopf algebra Show that its category of right comodules is braided via 'I'(v®w) _ ®v(1) R(v(2) ®w(2)) 4.* Let B be a braided group in a braided category Show that if V, W are B-modules in the category, then so is V W as claimed in Lecture 16 Define a braided B-module algebra as an algebra A in the category and a B-module such that the product (and unit) maps are morphisms Show that the adjoint action Ad makes B a B-module algebra (Hint: use diagrams.) What is the braided adjoint action of the braided line A' = k[x] on itself, in the category of kqZ-comodules? Here the coaction is x H x ® g where kqZ = k[g, g-1] Show that the braided coproduct of the quantum plane A42 is necessarily m A(xmyn) = E n r=o s=o [m] r q2 [n] s q2 xrys ®xm-ryn sq(m-r)s on general basis elements (Hint: obtain Oxen and Dyn first and then consider the braiding 1(xm-r ®ys) implied by W(x ®y) = qy ®x and functoriality under the product.) Hence or otherwise, obtain the partial derivatives aq,x and 8q,y stated in Lecture 15, and check that aq,yaq,x = q-'aq,xaq,y 7.* Check that if B is a braided group with invertible antipode then www.pdfgrip.com Problems 163 BOoP with coproduct T' o A is a braided group in the category with opposite braiding (Hint: try S°°P = S-1.) If A is an H-module algebra, show that A>iH with product (a® h) (b ®g) = a(h(l) 'b) ®h(2)g for a, b E A and h, g E H is associative If C is a right H-comodule coalgebra, show that is a coalgebra in a natural way 9.* If H is dual quasitriangular, check that there is a monoidal functor MH MH given by (V,,3) H (V, 0,,i), vah = O)R(v(2), h), Vv E V, h E H 10 Let q be a primitive n'th root of Compute the bosonisation of k[x]/(xn) in the category of kgZ1n-modules, where g'x = qx 11 Let k[y]/(y') be the braided group in the same category as in Q10, where g'y = q-1y Show that ev(y (9 x) = extends as pairing of braided groups by ev(y' ®xb) = Sa,b[a]q! What is the corresponding coevaluation coev (so that k[x]/(xn) is rigid)? www.pdfgrip.com Problems 164 Problems III * questions are optional For f E Aq = k[x] the braided line (in the category kqZ-comodules), prove the braided Taylor's theorem ega9 f (y) = f (x + y) in the braided tensor product algebra k[x]®k[y] Here 8q act in the second factor (the y variable) and not on x (Hint: the right hand side isAf.) A Hopf *-algebra is a Hopf algebra over C which is a *-algebra (i.e equipped with an antilinear antialgebra map obeying *2 = id) such that * commutes with A, c o * = - o e (where - is complex conjugation) and (S o *)2 = id In the finite-dimensional case, show that H* is a Hopf *-algebra with (0*, h) = (0, (Sh)*) for all h c H, E H* Check that the following are Hopf *-algebra structures on Uq(sl2) over C (i) q real, g* = g, E* = gF, F* = Eg-1 (this is called Uq(su2)) (ii) q real, g* = g, E* _ -gF, F* = -Eg-1 (this is called Uq(sui,i)) (iii) q of modulus 1, g* = g, E* = -qE, F* = -q-1F (this is called Uq(s12(I8))) 4.* Let H be a Hopf algebra Check that D(H)=H`PmH by mutual coadjoint actions ha¢ _ (Sha>)h(3>), h E H and E H* Here S denotes the antipode of H or H* Let (g, 5) be a finite-dimensional Lie bialgebra Check that g* is also, with the dual structure maps Find the dual of the Lie bialgebra s12 defined by [H, X+] = ±2Xt, 6(H)=O, [X+, X-] = H, 5(Xf) = (X f ®H - H ®X+) 6.* Check that the fixed subalgebra of SLq(2) under the coaction k[g,g-1] is the algebra of the q-sphere given in the Lecture 23 www.pdfgrip.com Problems 165 7.* The quantum differential calculus associated to the field extension R C C is 1l = R[x] and 1' = C[x] Considering the 1-forms dx and w = xdx - (dx)x as a basis of C[x] as a right 1[8[x]-module, show that the left module structure is xdx = (dx)x + w, xw = wx - dx Obtain an explicit description for the exterior derivative d : 1[8[x] -* C[x] www.pdfgrip.com Bibliography T Brzezinski and S Majid, Quantum group gauge theory on quantum spaces, Commun Math Phys 157 (1993) 591-638 V.G Drinfeld, Quantum Groups, in Proceedings of the ICM, A.M.S 1987 G Lusztig, Introduction to Quantum Groups, Birkhaser 1993 S Mac Lane, Categories for the Working Mathematician, Springer-Verlag 1974 S Majid, Algebras and Hopf Algebras in Braided Categories, Lee Notes Pure Appl Math 158 (1994) 55-105 Marcel Dekker S Majid, Quantum and braided Lie algebras, J Geom Phys 13 (1994) 307-356 S Majid, Foundations of Quantum Group Theory, C.U.P 1995 S Majid, Quantum and braided diffeomorphism groups, J Geom Phys 28 (1998) 94-128 S Majid, Classification of bicovariant differential calculi, J Geom Phys 25 (1998) 119-140 S Majid, Double bosonisation of braided groups and the construction of UQ(g), Math Proc Camb Phil Soc 125 (1999) 151-192 M.E Sweedler, Hopf Algebras, Benjamin 1969 S.L Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Commun Math Phys 122 (1989) 125-170 166 www.pdfgrip.com Index braided antipode, 86 braided category, 54, 66 braided coregular action, 92 braided derivative, 94, 115, 122, 130 braided determinant, 96 braided dimension, 72 braided enveloping algebra, 88 braided group, 86 braided integer, 130 braided-Lie algebra, 88, 131, 158 braided line, 89 braided matrices, 97, 107 braided module, 100 braided plane, 89, 95, 103, 113, 128 braided tensor product, 97, 133 braided tensor product algebra, 85 braided trace, 72 A(R), 51, 67, 126 BSLq(2), 96 GLq(2), 22 kqZ, 49, 66, 89 kqZ/, , 35, 59, 75, 102 MH, 66 HM, 58 Mq(2), 22, 127 Sl2i 141 sl2, 142 SLq(2), 10 uq(b+), 102 Uq(b+), 5, 102 Uq(g), 113, 119, 124 Uq(re+), 117, 120, 124 uq(s12), 36, 111, 114 Uq(S12), 41, 64 braiding, 55 action, 18 adjoint action, 14 adjoint coaction, 20 algebra, algebra factorisation, 133 algebraic group, 10 antialgebra map, anticoalgebra map, antipode, 4, 34 anyonic line, 102 automorphism, 14, 23 bialgebra, bialgebra factorisation, 134 bicharacter, 31, 48 bicovariant differentials, 156 bicrossproduct, 136, 144 bimodule, 149 bosonisation, 100 braid group, 58 braided additive coproduct, 128 braided adjoint action, 87 canonical braiding, 62, 68 canonical connection, 154 Cartan datum, 115 classical double, 142 classical Yang-Baxter equation, 140 coaction, 18 coadjoint action, 17 coalgebra, cocommutative, cocycle, 56, 136 coevaluation, 69, 70 comeasuring bialgebra, 23 commutative, comodule, 19 comodule algebra, 19 comodule category, 66, 103 comodule coalgebra, 19 conjugacy class, 158 conjugate quasitriangular, 30 connection, 153 contravariant functor, 52 167 www.pdfgrip.com Index 168 Hopf algebra, Hopf fibration, 152 Hopf-Galois extension, 149 convolution algebra, 47 coproduct, coquasitriangular, 47 coregular action, 15 counit, cross coproduct, 100 cross product, 99 crossed bimodules, 107 crossed G-module, 61 crossed module, 61, 156 cyclic group, 35 induced crossed module, 98 inhomogeneous quantum group, 103 irreducible calculus, 157 irreducible representation, 64, 158 isotopic, 77 Iwasawa decomposition, 144 Jones polynomial, 82 derivation, 14 diagrammatic notation, 55, 84 diffeomorphism, 23 differential calculus, 149 double bosonisation, 109 double cross product, 132, 134 double cross sum, 143 Drinfeld theory, 29 Drinfeld-Sklyanin solution, 141 dual braided group, 92, 108 dual Lie bialgebra, 139 dual object, 69 dual quantum group, 12 dual quasitriangular, 47, 66, 127 dual representation, 68 dually paired, 12, 118, 122 Dynkin classification, 112 enveloping algebra, 124 evaluation, 69, 70 extension of bialgebras, 136 exterior derivative, 149 factorisable, 31, 35, 41, 45, 49, 59, 107 fermionic quantum plane, 129 field extension, 28, 157 finite group, 6, 9, 31, 56, 60, 63, 132, 137, 158 formal powerseries, 42, 140 framed link, 80 free algebra, 117 functor, 52 fundamental group of link, 81 generalised braiding, 133 group factorisation, 132 group function algebra, 9, 30 group-graded, 57, 60 group Hopf algebra, grouplike, half-real form, 143 Hecke algebra, 67, 68 hexagon condition, 54 highest weight, 64, 75 Killing form, 31, 124, 141 left dual, 79 Lie algebra, 7, 63, 88, 139 Lie algebra splitting, 142 Lie bialgebra, 139 link diagram, 77 link invariant, 79 Lusztig formulation, 115 matched pair, 132 matrix coproduct, 10, 21, 27, 50, 96, 126 module algebra, 13, 114, 124 module category, 58, 98 module coalgebra, 17 monoidal category, 53 monoidal functor, 98 morphism, 52 natural transformation, 52 Newton, Isaac, 158 nonuniversal differential, 155 normalisation factor, 95 object, 52 octonions, 57 pentagon condition, 53 Peter-Weyl decomposition, 75 Poisson manifold, 146 Poisson-Lie group, 147 push out, 24 q-analysis, 39 q-binomial, 6, 39, 122 q-derivative, 16, 94, 115 q-determinant, 11 q-exponential, 40, 125 q-Hecke, 129 q-integers, 6, 39, 75 q-monopole, 155 q-Serre relation, 122 q-sphere, 151 quantisation, 149 www.pdfgrip.com Index quantum dimension, 75 quantum double, 44, 60, 62, 106, 135 quantum homogeneous space, 154 quantum matrix, 22, 50, 67, 126, 152 quantum order, 75 quantum plane, 21, 89, 103, 113, 128 quantum principal bundle, 149 quasicocommutative, 29, 59 quasicommutative, 47 quasi-Hopf algebra, 56 quasitriangular, 29, 45, 58, 110, 140 R-matrix, 50, 66, 126, 148 Reidemeister move, 77 ribbon, 34, 41, 73 right dual, 79 rigid category, 69 root datum, 115 superspace, 60, 75 symmetric category, 55, 60 Tannaka-Krein reconstruction, 105 tensor Hopf algebra, tensor product action, 13 tensor product algebra, tensor product coaction, 19, 66 theta-function, 35, 75 transmutation, 105 trefoil dimension, 82 triangle condition, 53 triangular, 31, 59 trigonometric bialgebra, 28 unit object, 54 universal differential, 150 universal enveloping algebra, 7, 63 Schur-Weyl duality, 68 second inverse, 126 self-dual, 12, 107 semisimple, 31, 34, 113, 118, 124 skein relation, 82 skew-derivation, 14 Verma module, 124 solvable Lie algebra, 144 Yang-Baxter equation, 30, 58, 140 Yang-Baxter variety, 131 superline, 102 weak quasitriangular structure, 116 Weyl group, 125 writhe, 78 www.pdfgrip.com 169 ... ordinary exponentials are related to the additive group R.) www.pdfgrip.com Coalgebras, bialgebras and Hopf algebras Uq(b+) 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... Hopf algebra means as a coalgebra So a bialgebra or Hopf algebra can either act as an algebra or coact as a coalgebra Similarly, the arrow-reversal of the notion of a module coalgebra in Figure... Definition 2.6 A bialgebra or Hopf algebra H acts on an algebra A (one says that A is an H-module algebra) if H acts on A as a vector space The product map m : A& A -> A commutes with the action of