1 This book is an introduction to geometric representation theory. 2 What is geometric representation theory? It is hard to define exactly 3 what it is as this subject is constantly growing in methods and scope. 4 The main aim of this area is to approach representation theory which 5 deals with symmetry and non-commutative structures by geometric 6 methods (and also get insights on the geometry from the representa- 7 tion theory). Here by geometry we mean any local to global situation 8 where one tries to understand complicated global structures by gluing 9 them from simple local structures. The main example is the Beilinson- 10 Bernstein localization theorem. This theorem essentially says that the 11 representation theory of a semi-simple Lie algebra (such as sl(n,C)) is 12 encoded in the geometry of its flag variety. This theorem enables the 13 transfer of “hard” (global) problems about the universal enveloping al- 14 gebra, to “easy” (local) problems in geometry. The Beilinson-Bernstein 15 localization theorem has been extremely useful in solving problems in 16 representation theory of semi-simple Lie algebras and in gaining deeper 17 insight into the structure of representation theory as a whole. There 18 are many more examples of geometric representation theory in action, 19 from Deligne-Lusztig varieties to the geometric Langlands’ program 20 and categorification. 21 The focus of this book is the Beilinson-Bernstein localization theo- 22 rem. It follows the advice of the great mathematician Israel M. Gelfand: 23 we only cover the case of sl2 (classical and quantum). This approach 24 allows us to introduce many topics in a very concrete way without go- 25 ing into the general theory. Thus we cover the Peter-Weyl theorem, 26 the Borel-Weil theorem, the Beilinson-Bernstein theorem and much 27 more for both the classical and quantum case. Dealing with the quan- 28 tum case allows us also to introduce many tools from non-commutative 29 algebraic geometry and quantum groups. These topics are usually con- 30 sidered very advanced. To have a full understanding of them requires 31 a good grasp of algebraic geometry, D-module theory, category theory, 32 homological algebra and the theory of semi-simple Lie algebras. We 33 think that by focusing on the simplest case of sl2 the student can gain 34 much insight and intuition into the subject. A good and deep under- 35 standing of sl2 makes the general theory much simpler to learn and 36 appreciate. 37 This book is based on a graduate lecture course given at MIT by 38 the second author. We are grateful to the students taking that course 39 for sharing their notes with us as we prepared this manuscript.

i 40 Contents

41 Introduction 1

42 Chapter 1. The first classical example: sl2. 2 43 1. The Lie algebra sl2 3 44 2. Irreducible finite-dimensional modules 4 45 3. The universal enveloping algebra 5 46 4. Semisimplicity 6 47 5. Characters 8 48 6. The PBW theorem, and the center ofU(sl 2) 9

49 Chapter 2. Hopf algebras and tensor categories 12 50 1. Hopf algebras 13 51 2. The first examples of Hopf Algebras 16 52 2.1. The Hopf algebraU(sl 2) 16 53 2.2. The Hopf algebra (SL ) 16 O 2 54 3. Tensor Categories 17

55 Chapter 3. Geometric Representation Theory forSL 2 20 56 1. The algebra of matrix coefficients 21 57 2. Peter-Weyl Theorem for SL(2) 22 58 3. Reconstructing (SL ) fromU(sl ) via matrix coefficients. 23 O 2 2 59 4. Equivariant vector bundles, and sheaves 24 60 5. Quasi-coherent sheaves on the flag variety 26 61 6. The Borel-Weil Theorem 28 62 7. Beilinson-Bernstein Localization 30 1 63 7.1.D-modules onP 30 64 7.2. The Localization Theorem 31

65 Chapter 4. The first quantum example:U q(sl2). 34 66 1. The quantum integers 35 67 2. The quantum enveloping algebraU q(sl2) 35 68 3. Representation theory forU q(sl2) 37 69 4.U q is a Hopf algebra 38 70 5. More representation theory forU q 39 71 5.1. Quantum Casimir element 40 ii CONTENTS iii

72 6. The locally finite part and the center ofU q(sl2) 41

73 Chapter 5. Categorical Commutativity in Braided Tensor 74 Categories 43 75 1. Braided and Symmetric Tensor Categories 44 76 2. R-matrix Preliminaries 46 77 3. Drinfeld’s UniversalR-matrix 47 78 4. Lusztig’sR-matrices 47 79 5. Weights of Type I, and bicharacters (needs better title) 48 80 6. The Yang-Baxter Equation 50 81 7. The hexagon Diagrams 53

82 Chapter 6. Geometric Representation Theory forSL q(2) 54 83 1. The Quantum Group Algebra 55 84 2. Generators and Relations forO q 55 85 3.O q comodules 57 86 4. Borel-Weil Correspondences in Classical and Quantum 87 Cases 58 88 4.1. The Quantum case 59 89 4.2. An equivalence of categories arising from the Hopf pairing 60 90 4.3. Restriction and Induction Functors 61 1 91 4.4. QuantumP 62 92 5. Lecture 14 - Quasi-coherent sheaves 64 93 5.1. Classical case 64 94 5.2. Quantum case 66 2 95 5.3. Quantum differential operators onA q 68 96 6. QuantumD-modules 70

97 Bibliography 75 INTRODUCTION 1

98 Introduction

99 In representation theory, the Lie algebrasl 2 :=sl 2(C) comprises the 100 first and most important example of a semi-simple Lie algebra. In this 101 introductory text, which grew out of a course taught by the first au- 102 thor, we will walk the reader through important concepts in geometric 103 representation theory, as well as their quantum group analogues. Our 104 focus is on developing concrete examples to illustrate the geometric 105 notions discussed in the text. As such, we will restrict our attention 106 almost exclusively to sl2, giving more general definitions only when it 107 is convenient or illustrative. 108 In Chapter 1, we show that the category of finite-dimensional sl2- 109 modules is a semi-simple abelian category; we prove this important fact 110 in a way which will generalize most easily to the quantum setting in 111 later chapters. 112 In Chapter 2, we introduce the formalisms of Hopf algebras and 113 tensor categories. These capture the essential properties of algebraic 114 groups, their representations, and their coordinate algebras, in a way 115 that can be extended to the quantum setting. 116 In Chapter 3, we discuss the relation between geometry of various 117 G-varieties and the representation theory ofG. We discuss the Peter- 118 Weyl theorem, and obtain as a corollary the Borel-Weil theorem. We 1 119 define D-modules onP , and we relate them to representations of sl2: 120 this is the first instance of the so-called Beilinson-Bernstein localization 121 theorem. 122 In Chapter 4, we introduce the quantized universal enveloping al- 123 gebraU q(sl2), and extend the results of Chapter 1 to the quantum 124 setting. 125 In Chapter 5, we explain the notion of a braided tensor category, 126 a mild generalization of the notion of a symmetric tensor category. 127 Braided tensor categories underlie the representation theory ofU q(sl2) 128 in a way analogous to the role of symmetric tensor categories in the 129 representation theory of sl2. 130 In Chapter 6, we reproduce the results of Chapter 3 in the quantum 131 setting. We have quantum analogs of each of the Peter-Weyl, Borel- 132 Weil, and Beilinson-Bernstein theorems. 133 Throughout the text assume some passing familiarity with the the- 134 ory of Lie algebras. Two excellent introductions are Humphreys [?] 135 and Knapp [?]. CHAPTER 1

136 The first classical example: sl2.

2 1. THE LIE ALGEBRA sl2 3

137 1. The Lie algebra sl2

The Lie algebrasl 2 :=sl 2(C) consists of the traceless 2 2 matrices, with the standard Lie bracket: × [A, B] :=AB BA. − 138 A standard and convenient presentation of sl2(C) is given as follows. 139 We let: 0 1 0 0 1 0 (1)E= ,F= ,H= , 0 0 1 0 0 1 � � � � � − � Then sl2 is spanned byE,F, andH, with commutators: [H,E]=2E,[H,F]= 2F,[E,F]=H. − 140 Recall that a representation ofg (equivalently, ag-module) is a vec- 141 tor spaceV , together with a Lie algebra homomorphismρ:g End(V). → 142 We will often omitρ from notation, and write simply x.v forρ(x).v. 143 The finite-dimensional representations ofsl 2 are sufficiently compli- 144 cated to be interesting, yet can be completely understood by elemen- 145 tary means. In this chapter, we recall their classification. We begin 146 with some examples:

147 Example 1.1. The defining representation. The Lie algebra sl2 2 148 acts onC by matrix multiplication.

149 Example 1.2. The adjoint representation. Any Lie algebrag acts 150 on itself by x.y:=[x, y].

151 Example 1.3. Given any representationV of a Lie algebrag, its 152 dual vector spaceV ∗ carries an action defined by (X.f)(v)=f( X.v). − 153 The corresponding representation is also denotedV ∗.

154 Example 1.4. Given two representationsV andW ofg, the vector 155 spaceV W carries an action ofg defined byx(v,w):=(xv,xw) for ⊕ 156 (v,w) V W, andx g. The corresponding representation is also ∈ ⊕ ∈ 157 denotedV W. ⊕ 158 Example 1.5. Given two representationsV andW , the vector 159 spaceV W carries an action ofg defined byx(v w) =x(v) ⊗ ⊗ ⊗ 160 w+v x(w), forv w V W , andx g. The corresponding ⊗ ⊗ ∈ ⊗ ∈ 161 representation is also denotedV W. ⊗ 162 As we will see in Chapter 2, these examples make the category of 163 g-modules into an abelian tensor category with duals (see also [?]). 2. IRREDUCIBLE FINITE-DIMENSIONAL MODULES 4

164 2. Irreducible finite-dimensional modules

165 Definition 2.1. LetV bean sl -module. A non-zerov V is 2 ∈ 166 a weight vector of weightλ ifHv= λv.A highest weight vector isa 167 weight vectorv ofV such thatEv = 0. Denote byV λ the subspace of 168 weight vectors of weightλ.

169 Observe that commutation relations (1) implyEV V , and λ ⊂ λ+2 170 FVλ V λ 2. ⊂ − 171 Exercise 2.2. Prove that every finite dimensional sl2 module has 172 a highest weight vector.

173 It follows that any irreducible finite dimensional representation is 174 generated by a highest weight vector; this fact will be the key to their 175 classification.

176 Lemma 2.3. Let V be a finite-dimensinal sl2-module, and suppose i 177 there exists a highest weight vectorv 0, of weightλ. Letv i := (1/i!)F (v0) 178 (by convention,v 1 = 0). Then we have that: − (2) Hvi = (λ 2i)v i, F vi = (i+1)v i+1, Evi = (λ i + 1)v i 1. − − − 179 Proof. The first two relations are obvious, and the third is a 180 straightforward computation:

iEvi = EFvi 1 = [E,F]v i 1 +FEv i 1 − − − = Hvi 1 +FEv i 1 = (λ 2i+2)v i 1 + (λ i+2)Fv i 2 − − − − − − = (λ 2i + 2)v i 1 + (i 1)(λ i + 2)v i 1 =i(λ i + 1)v i 1. − − − − − − − 181 �

182 Theorem 2.4. LetV be an irreducible finite dimensionalsl 2-module. 183 ThenV has a unique (up to scalar) highest weight vector of weight 184 m := dimV 1. Further,V decomposes as a direct sum of one dimen- − 185 sional weight spaces of weights m, m 2,...,2 m, m. − − − 186 Proof. It follows from Lemma 2.3 that span vi i N is a submodule { } ∈ 187 ofV , and thus all ofV . We letm 0 be maximal such thatv = ≥ m � 188 0 (equivalently,v m+1 is the first which is zero). Then by the third 189 equation of equation (2): 0 = Ev = (λ m)v . Therefore we m+1 − m 190 see thatλ=m, and that dimV=m + 1. Further, it is immediate 191 that the three formulas (withλ=m) define a representation of sl 2 192 on a vector space of dimensionm + 1, which we will denoteV(m). 193 Any such representation is irreducible, as applyingE to a vectorw 194 repeatedly will eventually yield a nonzero multiple ofv 0, and thusw 195 generates all ofV. � 3. THE UNIVERSAL ENVELOPING ALGEBRA 5

196 We note three important examples: firstly, the trivial representation 197 is the weight zero irreducible. The defining representation of sl2 on 2- 198 space is the weight one irreducible. Finally, we note that the adjoint 199 representation is three dimensional of highest weight 2, and this implies 200 that sl2 is a simple Lie algebra.

201 3. The universal enveloping algebra The universal enveloping algebraU(g), of a Lie algebrag, is the quotient of the free associative algebra on the vector spaceg (i.e. the tensor algebraT(g)), by the commutator relationsa b b a=[a, b]. That is, ⊗ − ⊗ U(g) :=T(g)/ a b b a [a, b] . � ⊗ − ⊗ − � 202 The canonical inclusiong� T(V ) induces a natural mapi:g → → 203 U(g). This gives rise to a functorU from Lie algebras to associative 204 algebras. We also have a forgetful functorF from associative algebras 205 to Lie algebras, given by defining [a, b] := ab ba, and then forgetting − 206 the associative multiplication.

207 Remark 3.1. Actually, the PBW theorem implies that the map 208 i:g U(g) is an inclusion, but this is not needed in what follows. → 209 Proposition 3.2. The functors(U, F) form an adjoint pair.

210 Proof. We need an isomorphismφ : Hom(U(g),A) Hom(g,F(A)). → 211 Givenf:U(g) A, we defineφ(f)=f i. It is easy to check that → ◦ 212 this gives the required isomorphism. �

213 By the adjointness above,ag-module is the same as an associative 214 algebra homomorphismρ:U(g) End(V ). In other words, we have → 215 an equivalence of categoriesg-Mod U(g)-Mod. Thus we may view ∼ 216 representation theory of Lie algebras as a sub-branch of representation 217 theory of associative algebras, rather than something entirely new. 218 The universal enveloping algebra of sl2 contains an important cen- 219 tral element, which will feature in the next section.

Definition 3.3. The Casimir element,C U(sl 2), is given by the formula: ∈ H2 C=EF+FE+ . 2

220 Claim 3.4.C is a central element ofU(sl 2). 4. SEMISIMPLICITY 6

Proof. It suffices to show that C commutes with the generators E,F,H. We compute: H2 [E,C]=[E,EF]+[E,FE]+[E, ] 2 = [E,E]F+E[E,F]+[E,F]E+F[E,E] 1 + ([E,H]H+H[E,H]) 2 =EH+HE EH HE=0. − − 221 The bracket [C,F ] is zero by a similar computation or by consideration 222 of the automorphism switchingE andF and takingH to H. Taking the bracket withH gives: − [H,C]=[H,E]F+E[H,F]+[H,F]E+F[H,E] = 2EF 2EF 2FE+2FE=0, − − 223 which proves the claim. �

224 4. Semisimplicity

225 Having classified irreducible finite dimensional representations, we 226 now wish to extend this classification to all finite dimensional repre- 227 sentations. This is accomplished by the following:

228 Theorem 4.1. The category of finite dimensional sl2-modules is 229 semisimple: any finite dimensional sl2-module is projective and thus 230 decomposes as a direct sum of simples.

231 In the proof of the theorem, we will use the following characteriza- 232 tion of semi-simplicity:

233 Exercise 4.2. Show that an abelian category is semi-simple if, and 234 only if, for every objectX the functor Hom(X, ) is projective. Hint: − 235 for the “if” direction, consider an exact sequence 0 U V W → → → → 236 0, and apply the functor Hom(W, ) to produce the required splitting − 237 W V. → By the exercise, we need to show that, for any finite dimensional

sl2-moduleX, the functor Hom sl2 (X, ) is exact on finite dimensional modules. We have a natural isomorphism,−

φ: Hom (V,W ∗ L) ∼ Hom (V W, L). sl2 ⊗ → sl2 ⊗ f φ(f), �→ 4. SEMISIMPLICITY 7

whereφ(f)(v w) := f(v), w . LetI denote the trivial representation; ⊗ � � we have a natural isomorphism,X ∼= I X, for anyX. Therefore we have natural isomorphisms: ⊗

Hom (X,V) = Hom (I X,V) = Hom (I,X∗ V). sl2 ∼ sl(2) ⊗ ∼ sl2 ⊗ 238 As these are all vector spaces, tensoring byX ∗ is an exact functor. So 239 we see that to prove the claim it suffices to show that Hom (I, ) is sl2 − 240 exact. 241 A homomorphism from the trivial module intoV is simply the 242 choice of a vectorv with the property that xv = 0 for allx sl 2. The sl ∈ 243 set of all suchv is a submodule ofV , denotedV 2 , which is naturally

244 isomorphic to Homsl2 (I,V ). So we have reduced the above theorem 245 to:

246 Lemma 4.3. For any finite dimensional sl2-moduleV , the functor sl 247 V V 2 is an exact functor. → 248 The proof of this lemma will rely upon the central Casimir element 249 C U(sl ). Note that, by Schur’s lemmaC will act as a scalar on any ∈ 2 250 finite dimensional irreducibleV.

251 Exercise 4.4. IfV is irreducible of highest weightm, thenC acts m2+2m 252 as scalar multiplication by 2 (hint: it suffices to compute the 253 action ofC on a highest-weight vector). k 254 Proposition 4.5. LetV a finite dimensional sl 2 module. IfC 255 acts as0 onV for somek>0, then sl 2 acts trivially onV.

256 Proof. We proceed by induction on dimV , the case dimV=0 257 being trivial. LetU V be a maximal proper submodule (U=0 ⊂ 258 is possible). By induction, sl2U = 0. Further, V/U is an irreducible 259 module, and by the above we know thatC acts as a nonzero scalar k 260 (and hence so doesC ) onV/U unless V/U is the trivial 1 dimensional 261 module. Thus, forv V, xv U for allx sl and so yxv=0 ∈ ∈ ∈ 2 262 for ally sl . Therefore [x, y]v = 0; however, since sl is a simple ∈ 2 2 263 Lie algebra, we have [sl2, sl2] = sl2, and thusV is a trivial module as 264 required. �

265 Remark 4.6.[?], [?] The Casimir elementC may defined for any 266 finite dimensional semi-simple Lie algebra, using the Killing form. It 267 can be shown that this is a central element which acts nontrivially on 268 nonzero irreducible modules.

269 Now, the following proposition finishes the argument:

270 Proposition 4.7. LetV a finite dimensional sl 2 module. Then 5. CHARACTERS 8

sl 271 (1) ker(C)=V 2 . 2 272 (2) ker(C ) ker(C). ⊆ 273 (3)V= ker(C) im(C). ⊕ sl 274 (4) The functorV V 2 is exact. �→ Proof. Claim (1) is immediate from Exercise 4.4 above; together with Proposition 4.5, it implies (2). We have ker(C) im(C)=0, by Claim (2), which implies (3). To see (4), we first∩ construct a chain complex V˜=0 V V 0, where the middle differen- tial is multiplication by→C→ (a morphism→ becauseC is central). We have ˜ ˜ sl2 H1(V)=H 0(V) ∼= V by (2). Suppose we have an exact sequence of sl2-modules 0 U V W 0. SinceC U(sl 2), the maps necessarily commute→ → with→ the→ differentials∈ to give an exact sequence of the complexes: j 0 U˜ i V˜ W˜ 0. → → → → We apply the snake lemma to obtain the long exact sequence, i j δ i j 0 U 1 V sl2 1 W sl2 U sl2 0 V sl2 0 W sl2 0. → 0 −→ −→ −→ −→ −→ → sl sl 275 Further, the induced mapi :U 2 V 2 may be identified with 0 → 276 the restriction of the original mapU W . By assumption this was → 277 injective, and so im(δ) = 0 and the induced right-hand sequence of 278 invariants is exact as required. � 279 Remark 4.8. The above proof can be slightly modified to apply to 280 a general semi-simple Lie algebra with Casimir elementC.

281 While Proposition 4.7 guarantees that a generalV can be split into 282 a direct sum of simple sl2 modules, the following is a more explicit 283 algorithm for constructing the decomposition.

284 (1) DecomposeV= V (m), whereV (m) denotes the eigenspace for ⊕ 2 285 the operatorC with eignevaluem + 2m k 286 (2) Within eachV , choose a basis v for theλ=m-weight (m) { i}i=1 287 space. 288 (3) SetV (m),i = sl2vi, which will be anm dimensional space by 289 our characterization above. 290 (4) ThenV= ( V ) is a decomposition into simple mod- ⊕ m ⊕i (m),i 291 ules.

292 5. Characters

293 The representation theory of sl2 admits a powerful theory of char- 294 acters, analogous to that of finite groups. Computing characters allows 295 us to easily determine the isomorphism type of any finite-dimensional 296 sl2-module, and to decompose tensor products. 6. THE PBW THEOREM, AND THE CENTER OFU(sl 2) 9

297 Definition 5.1. For a finite dimensional sl2 moduleV , we define 298 the formal sum: k ch(V)= (dimV k)x , k Z �∈ where we recall thatV k denotes the weight space, V = v V Hv=kv . k { ∈ | } Exercise 5.2. Definingx k x l =x k+l, we have: · ch(A B)= ch(A) + ch(B), ch(A B)= ch(A)ch(B). ⊕ ⊗ Example 5.3. By Theorem 2.4, we have: n+1 n 1 x x − − n n 2 2 n n ch(V(n)) = − =x +x − + +x − +x − . x x 1 ··· − − 299 Remark 5.4. SupposeV is a finite-dimensional sl 2-module,with 1 300 character ch(V ). Thenp(x)= ch(V) (x x − ) is a Laurent polynomial k · − 301 inx. The coefficient ofx inp(x) is the multiplicity of the irrreducible 302 V(k) inV.

303 Exercise 5.5. (Clebsch-Gordan) Give a decomposition ofV(m) ⊗ 304 V(n) as a sum of irreduciblesV(i) in two different ways: 305 (1) by finding all the highest weight vectors in the tensor product. 306 (2) by computing the character. n n 307 Exercise 5.6. Show that the subspace Sym (V(1)) ofV (1) ⊗ , 308 consisting of symmetric tensors, is a sub-module for thesl 2 action, and 309 is isomorphic toV(n).

310 The exercise implies that, as a tensor category, the category of 311 sl2-modules is generated by the objectV (1): in other words, every 312 irreducible sl2-module can be found in some tensor power ofV (1).

313 Exercise 5.7. Show thatV (1) V (1) = V (2) V (0). ⊗ ∼ ⊕ 314 We will see in next chapter that this is in some sense the only 315 relation in this category.

316 6. The PBW theorem, and the center ofU(sl 2)

317 The Poincare-Birkhoff-Witt theorem gives a basis ofU(g) for any 318 Lie algebrag. The proof we present hinges on a technical result in 319 non-commutative algebra known as the diamond lemma, which is of 320 independent interest. 321 Letk X denote the free algebra on a finite setX. Fix a total � � 322 ordering

324 a finite setS of pairs (m , f ), of a monomialm ink X , and a general i i i � � 325 elementf k X all of whose monomials are less thanm , or of smaller i ∈ � � i 326 degree. A general monomial ink X is called a PBW monomial if it � � 327 contains nom as a subword. A general element ofk X is called i � � 328 PBW-ordered if it is a sum of PBW monomials.

329 Lemma 6.1 (Diamond lemma, [?]). Suppose that: 330 (1) “Overlap ambiguities are resolvable”: For every triple of mono- 331 mialsA,B,C, with somem i =AB, andm j =BC, the expres- 332 sionsf iC and Afj can be further resolved to the same PBW- 333 ordered expression. 334 (2) “Inclusion ambiguities are resolvable”: For everyA,B,C, with 335 mi =B, andm j =ABC, the expressions Af iC andf j can be 336 further resolved to the same PBW-ordered expression. 337 Then, the set of PBW monomials ink X forms a basis for the quotient � � 338 ringk X / m f (m , f ) S . � � � i − i| i i ∈ � The defining relations ofU(sl 2) fit into the above formalism, with E < H < F and: S= (FE,EF H),(HE,EH+2E),(FH,HF+2F) . { − } 339 Theorem 6.2 (PBW Theorem). A basis forU(sl 2) is given by the k l m 340 PBW monomialsE H F , for k,l,m Z 0. ∈ ≥ Proof. We have only to check conditions (1) and (2) from Lemma 6.1. However, (2) is trivially satisfied, since the defining relations are at most quadratic in the generators. In fact, there is only one possible instance of condition (1), which is the monomialFHE. We compute: (FH)E=H(FE)+2FE=(HE)F H 2 + 2EF 2H − − =EHF+2EF H 2 + 2EF 2H. − − F(HE)=(FE)H+2(FE)=E(FH) H 2 + 2EF 2H − − =EHF+2EF H 2 + 2EF 2H. − − 341 �

342 Remark 6.3. In fact, with only slightly more effort, the diamond 343 lemma and the Jacobi identity together imply a related PBW theorem 344 for any Lie algebra - not necessarily semi-simple - over any field.

Corollary 6.4. We have an isomorphism of sl2-modules, k U(sl 2) ∼= Sym(sl2) := Sym (sl2). k 0 �≥ 6. THE PBW THEOREM, AND THE CENTER OFU(sl 2) 11

Proof. Define a filtration, •, ofsl 2-modules onU(sl 2) by declar- ing each ofE,H,F to be of degreeF one. Then it follows from Theorem 6.2 that the associated graded algebra, k k 1 grU(sl 2) = k 0 U(sl 2)/ − U(sl 2), ⊕ ≥ F F 345 is isomorphic to the symmetric algebra, Sym(sl2). However, since each 346 • is a finite-dimensional sl -module, and hence semi-simple, we have F 2 347 an isomorphismU(sl 2) ∼= grU(sl 2). �

Corollary 6.5 (Harish-Chandra isomorphism). The center ofU(sl 2) is freely generated by the Casimir element. We have an isomorphism:

ZU(sl 2) ∼= C[C]. 348 Proof. We present an elementary proof, which highlights the tech- 349 nique of characters. First, it is clear that the powers ofC are linearly k k k 350 independent, as the leading order PBW monomial ofC isE F . What 351 remains to show is that there are no other central elements. We note sl2 352 thatZU(sl 2) may be identified with the space of invariantsU(sl 2) : 353 forz U(sl ), we have [X,z] = 0 for allX if, and only if,z lies in the ∈ 2 354 center. Follwoing Corollary 6.4, let us define a weighted character ofU(sl 2) as follows: k k ch(U(sl 2)) := t ch(Sym V (2)). �k As aC[H]-module,� we haveV (2) = V 2 V 0 V 2, which implies an ∼ − ⊕ ⊕ isomorphism ofC[H]-modules,

Sym(V (2)) = Sym(V 2) Sym(V 0) Sym(V 2). ∼ − ⊗ ⊗ Thus, we have: 1 ch(U(sl )) = . 2 (1 x 2t)(1 t)(1 x 2t) − − − − The multiplicity� ofV (0) in each Sym kV(2) is the xt k coefficient of 1 p(x,t) = ch(U(sl )) (x x − ), following Remark 5.4. We have: 2 · − 1 x x − p(x,t) = − � (1 t)(1 x 2t)(1 x 2t) − − − − 1 x x 1 = − , 1 t 2 1 x 2t − 1 x 2t − � − − − � 1 355 which hasx-coefficient 1 t2 . It follows that there are no invariants in k− 2k 356 odd degrees, and thatC spans Sym (sl2), as desired.� CHAPTER 2

357 Hopf algebras and tensor categories

12 1. HOPF ALGEBRAS 13

358 1. Hopf algebras

359 In Example 1.4 of Chapter 1, for anyg-modulesV andW , we 360 endowed the vector spaceV W with ag-module structure. In this ⊗ 361 section, we consider a general class of associative algebras called Hopf 362 algebras, which come equipped with a natural tensor product operation 363 on their categories of modules. The enveloping algebraU(sl 2) will be 364 our first example. To begin, let us re-phrase the axioms for an algebra 365 in a convenient categorical fashion.

366 Definition 1.1. An algebra overC is a vector spaceA equipped 367 with a multiplicationµ:A A A, and a unitη:C A, such that ⊗ → → 368 the following diagrams commute:

η id id η µ id C A ⊗ � A A � ⊗ A C A A A ⊗ � A A ⊗ �� ⊗ ⊗ ⊗ ⊗ ⊗ �� ��� �� µ �� id µ µ �� �� ⊗ ∼= �� �� ∼= �� � � � � µ � A A A � A ⊗ 369 These diagrams represent the unit and associativity axoims, respec- 370 tively. Example 1.2. Given any two algebrasA andB, we can define an algebra structure on the vector spaceA B by the composition ⊗

id τ id µA µB A B A B ⊗ ⊗ � A A B B ⊗ � A B, ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 371 whereτ flips tensor components:τ(v w)=w v. ⊗ ⊗ 372 We define a co-algebra by dualizing the above notions (i.e. by 373 reversing all the arrows).

374 Definition 1.3. A co-algebra overC is a vector spaceA equipped 375 with a co-multiplication Δ :A A A, and a co-unit�:A C, such → ⊗ → 376 that the following diagrams commute.

� id id � Δ id C A � ⊗ A A ⊗ � A C A A AA� ⊗ A ⊗ � � ⊗� � ⊗ ⊗ ⊗� ⊗ � �� ��� �� Δ �� id Δ Δ �� �� ⊗ ∼= �� �� ∼= � �� Δ A A A � A ⊗ 377 By analogy, these are called the co-unit and co-associativity axioms, 378 respectively. 1. HOPF ALGEBRAS 14

Remark 1.4. For any co-algebraA,A ∗ becomes an algebra, via the composition

Δ∗ µ:A ∗ A ∗ � (A A) ∗ A ∗ ⊗ → ⊗ −→ of the natural inclusion , and the dual to the comultiplication map. if A is a finite-dimensional algebra, thenA ∗ becomes a co-algebra, via the composition,

µ∗ Δ:A (A A) ∗ = A∗ A ∗. −→ ⊗ ∼ ⊗ 379 However, forA infinite dimensional, this prescription does not lead to 380 a comultiplication map forA ∗, since the inclusionA ∗ A ∗ � (A A) ∗ ⊗ → ⊗ 381 is not an isomorphism. In the next chapter we’ll see a way around this 382 difficulty. Example 1.5. Given two co-algebrasA andB, we can define a co-algebra structure on vector spaceA B by ⊗ Δ Δ id τ id A B ⊗ � A A B B ⊗ ⊗ � A B A B ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 383 Definition 1.6.A bi-algebra is a vector spaceA equipped with 384 algebra structure (A, µ, η) and co-algebra structure (A,Δ,�) satisying 385 either of the conditions:

386 (1) Δ and� are algebra morphisms. 387 (2)µ andη are co-algebra morphisms

388 Exercise 1.7. Prove that (1) and (2) are equivalent (hint: write 389 out the appropriate diagrams, and turn your head to one side).

390 Exercise 1.8. Group algebras. LetG be a finite group, and let 391 C[G] denote its group algebra. Check that Δ(g)=g g and�(g)=δ e,g ⊗ 392 defines a bi-algebra structure onC[G],

393 Exercise 1.9. Enveloping algebra. Letg be a Lie algebra, and 394 U(g) its universal enveloping algebra. ForX g, define Δ(X) = ∈ 395 X 1+1 X, and�(X) = 0. Show that this defines a bi-algebra ⊗ ⊗ 396 structure onU(g).

397 Exercise 1.10. LetG be an affine algebraic group, and denote its 398 coordinate algebra (G). Define Δ(f) (G) (G) = (G G) O ∈O ⊗O ∼ O × 399 by Δ(f)(x y)=f(x y), where “ ” is the multiplication in the group. ⊗ · · 400 Define�(f) as projection onto the constant term. Show that this defines 401 a bi-algebra structure. You will need to show that Δ(f) is a polynomial 402 inx andy. 1. HOPF ALGEBRAS 15

403 Exercise 1.11. LetH be a bialgebra, and letI H be an ideal ⊂ 404 (with respect to the algebra structure) such that Δ(I) H I+I H ⊂ ⊗ ⊗ 405 (i.e.I is a co-ideal). Show that Δ and� descend, to form a bi-algebra 406 structure onH/I. Definition 1.12. LetA be a co-algebra,B an algebra. Let f,g: A B be linear maps. We define the convolution productf g as the composition:→ ∗

Δ f g µ A � A A ⊗ � B B � B ⊗ ⊗ 407 IfA is a bialgebra, then takingB=A above yields the structure 408 of an associative algebra on End(A), with unitη �. ◦ 409 Definition 1.13.A Hopf algebra is a bi-algebraH such that there 410 exists an inverseS:H H to Id relative to : that is, we have → ∗ 411 S id= id S=η �.S is called the antipode. ∗ ∗ ◦ 412 Remark 1.14. Note that the antipode on a bi-algebra is unique, if 413 it exists, by uniqueness of inverses in the associative algebra End(A).

414 The best way to understand the antipode is as a sort of linearized 415 inverse, as the following examples illustrate.

416 Exercise 1.15. DefineS for Examples 1.8, 1.9, 1.10, and show 417 that it defines a Hopf algebra in each case.

418 Exercise 1.16.(??, III.3.4) In any Hopf algebra,S(xy)=S(y)S(x). 419 Hint: Define ν,ρ Hom(H H,H) byν(x y) =S(y)S(x), and ∈ ⊗ ⊗ 420 ρ(x y)=S(xy). Then computeρ µ=µ ν=η �. ⊗ ∗ ∗ ◦ 421 Remark 1.17. In the case thatS is invertible, it is an anti-automorphism 422 and thus can be used to interchange the category of left and right mod- 423 ules overH.

424 Exercise 1.18. Suppose that the Hopf algebraH is either commu- 2 425 tative, or co-commutative. Show by direct computation thatS S= ∗ 426 η τ, and thus conclude thatS is an involution. ◦ Definition 1.19. For any bi-algebraH, andH-modulesM and N, we define their tensor productM N to have as underlying vector ⊗ spaces the usual tensor product overC, withH-action defined by:

Δ id τ23 µM µN H (M N) ⊗ � H H M N � H M H N ⊗ � M N ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 427 Exercise 1.20. Check thatM N is in fact anH-module, by ⊗ 428 verifying the associativity and unit axioms. 2. THE FIRST EXAMPLES OF HOPF ALGEBRAS 16

429 Exercise 1.21. Similarly, given twoH-comodulesM andN, we 430 can define a comodule structure on their tensor product. Define the 431 action, and check that it gives a well-defined co-module structure.

432 Remark 1.22. In Examples 1.8, 1.9, 1.10, we recover in this way 433 the usual action onM N. For instance ifG is a group, then in ⊗ 434 k[G], we haveg(v w) =g(v) g(w); ifg is a Lie algebra, we have ⊗ ⊗ 435 x(v w)=x(v) w+v x(w). ⊗ ⊗ ⊗

436 2. The first examples of Hopf Algebras

437 2.1. The Hopf algebraU(sl 2). We have previously definedU= 438 U(sl 2) as an algebra; by Example 1.9, we can endow it with a co- 439 product structure such that Δ(E)=E 1+1 E, Δ(F)=F 1+1 F ⊗ ⊗ ⊗ ⊗ Δ(H)=H 1+1 H, �(E)=�(F)=�(H)=0. ⊗ ⊗ Following Exercise 1.15,U has antipode given by: S(E)= E, S(F)= F, S(H)= H. − − − 2.2. The Hopf algebra (SL ). The algebraic group O 2 a b SL2 =SL 2(C)= a, b, c, d C, ad bc=1 c d | ∈ − �� � � has coordinate algebra (SL 2) :=C[a, b, c, d]/ ad bc 1 . We define a co-product for = (OSL ) on generators as� follows:− − � O O 2 Δ(a)=a a+b c, Δ(b)=a b+b d, ⊗ ⊗ ⊗ ⊗ Δ(c)=c a+d c, Δ(d)=c b+d d. ⊗ ⊗ ⊗ ⊗ We may write this more concisely as follows: Δ(a) Δ(b) a b a b = . Δ(c) Δ(d) c d ⊗ c d � � � � � � 440 Exercise 2.1. Let Δ:C[a, b, c, d] C[a, b, c, d] C[a, b, c, d] be → ⊗ 441 given by the formulas for Δ above. Show that:

442 (1) Δ(ad bc)=(ad bc) (ad bc), so that − − ⊗ − 443 (2) Δ(ad bc 1) (ad bc 1) H+H (ad bc 1). − − ⊂ − − ⊗ ⊗ − − 444 Conclude that Δ descends to a homomorphism Δ : . O→O⊗O 3. TENSOR CATEGORIES 17

This makes (SL 2) into a bi-algebra. We now introduce an an- tipode, which willO endow it with the structure of a Hopf algebra. We defineS on generators: S(a)S(b) d b = . S(c)S(d) c− a � � � − � 445 Exercise 2.2. Verify thatS is an antipode.

446 3. Tensor Categories

447 In the previous section, we saw that for any Hopf algebraH, the 448 category ofH-modules has a tensor product structure. In this section, 449 we will define the notion of a tensor category, which captures this prod- 450 uct structure. The reason for the focus on categorical constructions is 451 that when we look at the quantum analogs of our classical objects, 452 much of the geometric intuition fades, while the categorical notions 453 remain largely intact. Definition 3.1. Let , be categories. Their product, , is the category whose objectsC D are pairs (V,W),V ob( ),W C×Dob( ), and whose morphisms are given by: ∈ C ∈ D

Mor((U, V),(U �,V �)) = Mor(U, U �) Mor(V,V �). × 454 Let be a functor : . This means that for each pair ⊗ ⊗ C×C→C 455 (U, V) , we have their tensor productU V , and for any maps ∈C×C ⊗ 456 f:U U �, g:V V �, we have a mapf g:U V U � V �. → → ⊗ ⊗ → ⊗ 457 Definition 3.2. An associativity constraint on is a natural iso- ⊗ 458 morphisma :(U V) W U (V W ) which satisfies the U,V,W ⊗ ⊗ → ⊗ ⊗ 459 Pentagon Axiom. ((A B) C) D � (A (B C)) D a 1 ⊗ ⊗ ⊗ A,B,C ⊗ ⊗ ⊗ ⊗ aA B,C,D ⊗ � (A B) (C D) aA,B C,D ⊗ ⊗ ⊗ ⊗ aA,B,C D ⊗ � 1 aB,C,D � A (B (C D))A ((⊗� B C) D) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 460 Remark 3.3. It is useful to think of the functor as a cate- ⊗ 461 gorified version of an associative product. Whereas in the theory of 462 groups or rings (or more generally, monoids) one encounters the iden- 463 tity (ab)c=a(bc) expressing associativity of multiplication, this is not 464 sensible for categories, as objects are rarely equal, but more often iso- 465 morphic (consider the example of tensor products of vector spaces). It 3. TENSOR CATEGORIES 18

466 is an exercise to show that the basic associative identity for monoids 467 implies that any two parenthesizations of the same word of arbitrary 468 length are equal. In tensor categories, we need to impose an equal- 469 ity of various associators on tensor products of quadruples of objects. 470 MacLane’s theorem [] asserts that this commutativity on 4-tuples im- 471 plies the analogous equality of associators forn-tuples, so that we may 472 omit parenthesizations going forward.

473 Definition 3.4. A unit for is a triple (I, l, r), whereI , and ⊗ ∈C 474 l:I U U andr:U I I are natural isomorphisms. ⊗ → ⊗ → Definition 3.5. A tensor category is a collection ( , , a, I, l, r) with a, I, l, r as above, such that we have the followingC commutative⊗ diagram

a (A I) B � A (I B) . ⊗ ⊗ �� �⊗ ⊗ ��� ��� ��� � �� A B ⊗ Definition 3.6. A tensor functorF:( , ) ( , ) is a pair (F,J) of a functorF: , and a naturalC isomorphism⊗ → D ⊗ C→D

J :FA FB ∼ F(A B), I ∼ F(I) A,B ⊗ −→ ⊗ −→

475 such that diagrams

F(A B) FC ⊗�� ⊗ �� ��� ��� ��� ��� (FA FB) �FC� F((A B) C) ⊗ ⊗ ⊗ ⊗ FA (FB� FC) F(A (B C� )) ⊗ ⊗�� ⊗ �⊗ ��� ��� ��� ��� FA F� (B C�) ⊗ ⊗ and

FA I � FA ⊗ �

� FA FI � F(A I) ⊗ ⊗

476 commute, as well as the similar diagram for right unit constraints. 3. TENSOR CATEGORIES 19

Definition 3.7. A tensor natural transformation between tensor functorsF andG is a natural transformationα:F G is such that → FA FB � F(A B) ⊗ ⊗

� � GA GB � G(A B) ⊗ ⊗ 477 commutes. 478

479 Definition 3.8.( , ) is strict if a, l, r are all equalities in the C ⊗ 480 category (meaning that the underlying objects are equal, and the mor- 481 phism is the identity). A tensor functorF=(F,J) is strict ifJ isan 482 equality andI=FI.

483 Remark 3.9. Most categories arising naturally in representation 484 theory are not strict categories, but we will see in chapter ?? by an 485 extension of MacLane’s coherence theorem that any tensor category is 486 tensor equivalent to a strict category. In chapter ??, we will see some 487 examples of strict tensor categories.

488 Example 3.10. CHAPTER 3

489 Geometric Representation Theory forSL 2

20 1. THE ALGEBRA OF MATRIX COEFFICIENTS 21

490 In this chapter we begin the study of geometric representation the- 491 ory, in which techniques from algebraic and differential geometry are 492 brought to bear on the representation theory of algebraic groups. We 493 focus on three main results: 494 (1) the Peter-Weyl Theorem, which states that the coordinate al- 495 gebra (G), viewed as a leftG G-module, contains one di- O × 496 rect summand End(V ) for every finite dimensional irreducible 497 moduleV ofG; 498 (2) the Borel-Weil theorem, which realizes finite-dimensional rep- 499 resentations of a semi-simple algebraic group geometrically as 500 sections of certain equivariant line bundles on the correspond- 501 ing flag variety; and 502 (3) the Beilinson-Bernstein localization theorem, which gives an 503 equivalence between the category ofD-modules on the flag 504 variety and the category ofU(g)-modules with trivial central 505 character.

506 As in the previous chapter, we will look toSL 2 for most of our exam- 507 ples.

508 1. The algebra of matrix coefficients The finite dimensional representations of a (possibly infinite dimen- sional) Hopf algebraH determine a natural subalgebra ofH ∗, called the algebra of matrix coefficients, which is naturally a Hopf algebra, thus overcoming the finiteness issues in Remark??. The dual vector spaceH ∗ carries an action ofH H, given by: ⊗ ((a b)φ)(x) :=φ(S(b)xa). ⊗ 509 Definition 1.1. The external tensor productV�W ofH-modules 510 V andW is theH H-module with underlying vector spaceV W, ⊗ ⊗ C 511 and action (u u )(v w) :=u v u w. 1 ⊗ 2 ⊗ 1 ⊗ 2 512 LetV be a finite dimensionalH-module. Forf V ∗, v V, V V∈ ∈ 513 the matrix coefficientsc f,v H ∗ are defubed byc f,v(u) :=f(u.v), for ∈ V 514 u H. The assignment (f,v) c f,v is bi-linear; we thus obtain a ∈ V �→ 515 linear mapc :V ∗ V H ∗. � → V W V W 516 Exercise 1.2. Show thatc f,vcg,w =c g⊗f,v w . ⊗ ⊗ 517 Exercise 1.3. Letφ:V W be a homomorphism ofH-modules. → W V 518 Show that, forv V,f W ∗, we havec =c . ∈ ∈ f,φv φ∗f,v 519 Definition 1.4. The algebra, , of matrix coefficients, is the linear O 520 subspace ofH ∗ spanned by thec f,v for all finite-dimensional.V. 2. PETER-WEYL THEOREM FOR SL(2) 22

Exercise 1.5. Conclude that isaH H-submodule ofH ∗, and thatc V is aH H-module map, byO showing,⊗ for a, b H: ⊗ ∈ (a b)c =c . ⊗ f,v bf,av 521 Exercise 1.6. Fix a basisv 1, . . . , vn forV , and letf 1, . . . , fn forV ∗ 522 be the dual basis. Verify that the representation mapρ:U gl(V) n → 523 sendsx to the matrix (c fi,vi (x))i,j=1, thus justifying the name “matrix 524 coefficient”.

525 Exercise 1.7. Suppose thatH is commutative, or co-commutative, 526 so that the tensor flipv w w v is a morphism ofH-modules. ⊗ �→ ⊗ 527 Show in this case that is commutative. O 528 Proposition 1.8. LetΔ:H ∗ (H H) ∗ denote the dual to the → ⊗ 529 multiplication map onH. Then we have Δ( ) (H H) ∗, O ⊂O⊗O⊂ ⊗ 530 and this endows with the structure of a Hopf algebra. O 531 Proof. For the first claim, it suffices to show that Δc , f,v ∈ O ⊗ O 532 for each finite-dimensionalV , eachf V ∗, andv V . Let v bea ∈ ∈ { i} 533 basis forV and f a dual basis forV ∗. The proof follows from the { i} 534 following exercise: n 535 Exercise 1.9. Show that Δ(c ) = c c , by checking f,v i=1 f,vi ⊗ fi,v 536 that this expression satisfies: Δ(c f,v),x y = c f,v, xy . � ⊗�� � � 537 Having defined the bi-algebra structure, the antipodeS is defined 538 by S(c ),x = c ,S(x) , forx H. � f,v � � f,v � ∈ 539 �

540 2. Peter-Weyl Theorem for SL(2)

541 Returning to the caseU=U(sl 2), we have the following description 542 of the algebra of matrix coefficients. O Theorem 2.1. (Peter-Weyl) LetV(n) denote the irreducible rep- resentation of sl2 of highest weightn. Then we have an isomorphism ofU U-modules: ⊗ ∞ = V(j) ∗ V(j), O ∼ � j=0 � Proof. We have a map ofU U-modules, ⊗ ∞ ∞ V(j) c : V(j) ∗ V(j) . � →O j=0 j=0 � � V(j) 543 Eachc is an injection: the kernel is a submodule of the irreducible V(j) 544 U U-moduleV(j) ∗ V(j), and eachc is clearly not identically ⊗ ⊗ 3. RECONSTRUCTING (SL 2) FROMU(sl 2) VIA MATRIX COEFFICIENTS. 23 O V(j) V(k) 545 zero. Moreover, the images ofc andc must intersect trivially, 546 forj=k, since these are non-isomorphic irreducible submodules. It� only remains to prove surjectivity; we need to show that isin fact contained in the sum of of the images of the mapsc V(i) . ForO this, letV be an arbitrary finite dimensional representation, and using the semi-simplicity proved in Chapter 1, writeV as a finite direct sum of irreducibles: N mi V ∼= V(i) ⊕ . i=0 � Letπ i,j andι i,j, respectively, denote the projection onto, and inclusion into, thejth copy ofV(i) in the sum. We clearly haveπ i,j∗ =ι i,j. Let f V ∗, v V . Then we may write: ∈ ∈

v= ιi,jvi,j, f= πk,l∗ fk,l, i,j � �k,l for some collection ofv V(i) andf V(i) ∗. Thus, we have: i,j ∈ k,l ∈ cV = cV = cV . f,v πk,l∗ fk,l,ιi,j vi,j fk,l,πk,lιi,j vi,j i,j,k,l� i,j,k,l� 547 We haveπ k,lιi,j = IdV(i) ifi=k, and 0 otherwise. Thus the right hand V(i) 548 side lies in the span of the images of the mapsc , as desired.�

549 Remark 2.2. Clearly, both the statement and proof of the Peter- 550 Weyl theorem apply mutatis mutandis for any semi-simple algebraic 551 groups.

552 3. Reconstructing (SL ) fromU(sl ) via matrix coefficients. O 2 2 1 2 Choose a basisv 1, v2 ofV (1), and letv , v denote the dual basis of i i V (1)∗. We use the notationc j :=c v vj . We denote byi 0 andπ 0 the maps: ⊗

i :V (0) V (1) V (1), π :V (1) ∗ V (1) ∗ V (0) 0 → ⊗ 0 ⊗ → 1 v v v v a vi v j (a a ) �→ 1 ⊗ 2 − 2 ⊗ 1 ij ⊗ �→ 12 − 21 Thusi 0 andπ 0 are the inclusion and� projection, respectively, of the trivial representation relative to the decomposition, V (1) V (1) = V (2) V (0). ⊗ ∼ ⊕ 553 Exercise 3.1. The purpose of this exercise is to construct an iso- 554 morphism between (SL ) and the algebra of matrix coefficients on O 2 O 555 U(sl 2). 4. EQUIVARIANT VECTOR BUNDLES, AND SHEAVES 24

(1) Show that there exists a unique homomorphism: φ:C[a, b, c, d] , →O (a, b, c, d) (c 1, c1, c2, c2). �→ 1 2 1 2 556 (2) Show thatφ is surjective, using the fact thatV (1) generates 557 the tensor category of sl2 modules. 1 2 558 (3) Show that the relationsc f,i0(v) =c π0(f),v , forf=v , v and 559 v=v , v , reduce to the single relation ad bc = 1. 1 2 − (4) The algebra (SL 2) =C[a, b, c, d]/ ad bc 1 admits a fil- O � − − � tration with generators a, b, c, d in degree one. LetF i denote theith filtration, and show thatF i/Fi 1 has a basis: − = a kdlcm k+l+m=i a kdlbm k+l+m=i , Bi { | }∪{ | } i+2 2 560 so that dimF i/Fi+1 = i =2 2 (i + 1) = (i+1) . (5) Show thatφ is a map of|B filtered| vector− spaces, where � � Fi( )= k iV(k) ∗ �V(k). O ⊕ ≤ 561 (6) Conclude thatφ is injective, and thus an isomorphism of al- 562 gebras.

563 Exercise 3.2. Show thatφ is a isomorphism of Hopf algebras, by 564 showing that it respects co-products.

565 Remark 3.3. This exercise is the easiest case of a very general the- 566 ory, called Tannaka-Krein Reconstruction, which gives a prescription 567 for recovering the coordinate algebra of a reductive algebraic group 568 (more generally, any Hopf algebra) from its category of finite dimen- 569 sional representations.

570 4. Equivariant vector bundles, and sheaves LetX be an algebraic variety overC, andG an algebraic group. Let us denote the multiplication map onG by mult: G G multG × −→ SupposeG acts onX, meaning that we have an algebraic morphism: G X act X × −→ which is associative: act (mult 1) = (act) (1 act):G G X X ◦ × ◦ × × × → 571 Definition 4.1.AG-equivariant vector bundle onX is a vector 572 bundleπ:V X, overX, together with an actionG V V → × → 573 commuting withπ, and restricting to a linear mapφ :V V of g,x x → gx 574 each fiber. 4. EQUIVARIANT VECTOR BUNDLES, AND SHEAVES 25

It follows that the mapsφ g,x are linear isomorphisms, and are as- sociative in the following sense: φ φ =φ . h,gx ◦ g,x hg,x We will now give a generalization of this definition to sheaves. Us- ing the multiplication, action and projection we can form three maps, d , d , d :G G X G X: 0 1 2 × × → × 1 d0(g1, g2, x) = (g2, g1− x), d1(g1, g2, x) = (g1g2, x),

d2(g1, g2, x) = (g1, x). 575 We also have the identity section froms:X G X,s(x)=(e,x), → × 576 and the projection proj:G X X, proj(g,x) =x. × → Definition 4.2.AG-equivariant sheaf onX isapair( ,θ), where is a sheaf onX andθ is an isomorphism, F F θ: proj ∗ act∗ F −→ F satisfying the cocycle and unit conditions:

d0∗θ d 2∗θ=d 1∗θ, s∗θ= id . ◦ F 577 Exercise 4.3. Prove that ifV is an equivariant vector bundle then 578 the locally free sheaf of sections ofV is an equivariant sheaf.

579 Exercise 4.4. Prove that ifV isaG-equivariant locally free sheaf 580 onX, then Spec X (V ), the associated vector bundle onX isaG- 581 equivariant vector bundle.

582 Remark 4.5. Note that the we can give this definition also in other 583 categories (topological, differentiable, analytic,...).

584 Suppose now thatX= Spec(A) is an affine variety andG= 585 Spec(H) is an affine algebraic group, so thatH is a commutative Hopf 586 algebra. The action ofG onX translates intoA being aH-comodule 587 algebra:

588 Definition 4.6. AnH-comodule algebraAisanH-comodule, and 589 an algebra, such that the multiplication mapm:A A A is a map ⊗ → 590 of comodules, whereA A is anH-module via tensor product. ⊗ H 591 Definition 4.7. The category ofH-equivariantA-modules has C A 592 as objectsH-comodulesM, equipped with a mapm:A M M ⊗ → 593 ofH-comodules, makingM into anA-module. The morphisms in this 594 category are the maps that commute with both theA-module structure 595 and theH-comodule structure. 5. QUASI-COHERENT SHEAVES ON THE FLAG VARIETY 26

596 Exercise 4.8. In the setup of the preceding paragraph, construct H 597 an equivalence between and the category ofG-equivariant sheaves C A 598 onX.

599 Exercise 4.9. Suppose thatG acts transitively onX. Show that 600 aG-equivariant sheaf is locally free (hint: produce an isomorphism on 601 stalks, ). F x → Fgx 602 Exercise 4.10. LetX=G, and letG act on itself by left multipli- 603 cation. Show that the category of quasi-coherentG-equivariant sheaves 604 of -modules is equivalent to the category of vector spaces. O G 605 Exercise 4.11. LetX= pt with the trivialG-action. Show { } 606 that the category ofG-equivariant sheaves onX is equivalent to the 607 category of representations ofG.

608 5. Quasi-coherent sheaves on the flag variety For any semi-simple algebraic group, the flag variety is a homoge- neous space, the quotient G/B ofG by its Borel subgroupB. In the caseG=SL 2, the Borel subgroupB is the set of upper-triangular matrices, a b B= 1 . 0a − � � 609 We may identifyB with the stabilizer of the line spanned by the first 610 basis vector; the orbit-stabilizer theorem then gives an identification of 1 611 G/B with the first projective spaceP . 612 While G/B is a projective variety – in particular, not affine – we can 613 nevertheless approach its category of quasi-coherent sheaves without 614 appeal to projective geometry, by describing quasi-coherent sheaves on 615 G/B asB-equivariant sheaves onG. This purely algebraic point of 616 view will most easily generalize to the quantum case considered in the 617 next chapter, where most of the geometry is necessarily expressed in 618 algebraic terms.

619 Definition 5.1. The category of quasi-coherent sheaves on the 620 coset space G/B, denoted oh(G/B), has as objects allB-equivariant QC 621 -modules onG. Morphisms in oh(G/B) are those which commute O QC 622 with both the action and the (B)-coaction. O O 623 Remark 5.2. It is a theorem due to [] that the flag variety is in 624 fact an algebraic variety, and that furthermore its category of quasi- 625 coherent sheaves is equivalent to the category we have defined above.

626 Remark 5.3. Because theG-action is transitive, we can identify 627 the fibers of the sheaf for allx G/B. ∈ 5. QUASI-COHERENT SHEAVES ON THE FLAG VARIETY 27

628 More generally, for any Hopf we have a Hopf algebraH. The next 629 lemma generalizes Exercise 4.10.

H 630 Proposition 5.4. Vect. C H ∼ co inv co inv 631 Proof. LetM − = m M Δm=m 1 , thenM M − { ∈ | ⊗ } �→ 632 defines a functorF: Vect. The assignmentV V H gives C H → �→ ⊗ 633 a functorG : Vect H . To finish the proof, we need to produce →C co inv co inv 634 natural isomorphismsM = H M − and (V H) − = V. ∼ ⊗ ⊗ ∼ � 635 SupposeH has a quotient Hopf algebraA. We define a category 636 as the category whose objects areH-modulesM with a right - ACH O 637 comodule and leftA-module structures, such thatH M M isan ⊗ → 638 A-comodule map andH-comodule map. Δ 639 Here we useH H H A H to giveH anA-comodule −→ ⊗ → ⊗ 640 structure.

641 Lemma 5.5. LeftA-modules. ACH ∼ 642 Proof. This is an easy extension of Proposition 5.4.�

643 SinceG=SL(2) is an affine algebraic variety, the quasi-coherent 644 sheaves onG are just the (SL(2)) modules. In this case, the Borel O 645 subgroup is the groupU of upper triangular matrices. Thus, we can 1 646 construct the category ofP -modules as the category of (SL(2)) mod- O 647 ulesM which have (U)-comodule action, such that (SL(2) M O O ⊗ → 648 M is both an (SL(2))-module map, and a (U)-comodule map. This O O 1 649 gives us our first description of quasi-coherent modules onP . 650 There is a second, less general, construction of quasi-coherent sheaves 1 651 onP , which will give us a more explicit description. We note thatU= 652 T�N, whereT ∼= C× is the group of diagonal matrices, andN ∼= C 653 is the group of unipotent matrices. Thus,SL(2)/U ∼= (SL(2)/N)/T. 1b a b a0 N= ,U= 1 ,T= 1 , a, b C. 0 1 0a − 0a − ∈ �� �� �� �� �� �� 2 2 2 654 Exercise 5.6.SL(2)/N = A , whereA = Spec(C[x, y]), andA 2 ∼ ◦ 2 ◦ 655 denotesA 0 . It may be helpful to think ofA as the space of based \{ } 2 ◦ 656 lines (l,v) 0=v l C . { | � ∈ ⊂ } 2 2 657 Now let us describe oh(A ). We first recall that sinceA is affine, 2 QC ◦ 658 oh(A ) =C[x,y]-modules. QC 659 Definition 5.7.AC[x, y] moduleM is torsion if for anym M, l l ∈ 660 there exists anl>> 0 s.t.x m=y m=0. 6. THE BOREL-WEIL THEOREM 28

2 2 661 We consider the restriction functor Res: oh(A ) oh(A ). QC →QC ◦ 662 This is clearly surjective, since we can always extend a sheaf by zero 663 off of an open set.

664 Lemma 5.8. Res(M) ∼= 0 if, and only if,M is a torsion sheaf. 2 2 665 Proof. LetM be a torsion sheaf onA . OnA y-axis ,x is \{2 } 666 invertible, soM is necessarily zero there. Likewise, onA x-axis , y \{ } 2 667 is invertible, soM is zero there. Since these two open sets coverA , ◦ 668 we can conclude that torsion sheaves are sent to zero under restriction. 669 Conversely, ifM x andM y are both zero, thenM is a torsion sheaf.� 2 670 We would like now to conclude that oh(A ) is the quotient of 2 QC ◦ 671 oh(A ) by the full subcategory consisting of torsion modules. In QC 672 order to say this, we must define what we mean by the quotient of 673 a category by a subcategory. This is naturally defined whenever the 674 categories are abelian, and the subcategory is full, and also closed with 675 respect to short exact sequences. These notions, and the quotient con- 676 struction, are explained in the appendix ?? on abelian categories. 2 677 Theorem 5.9. oh(A ) C[x, y] modules/torsion. QC ◦ � − 1 678 Theorem 5.10. oh(P ) = gradedC[x, y] modules/torsion. QC − 679 Proof. TheC ∗ action onC[x, y] is dilation of each homogeneous deg(p) 680 component,λ(p(x,y)) =λ p(x,y). Thus, an equivariant module 681 with respect to this action inherits a gradingM k = m M λ(m) = k { ∈ | 682 λ m . Conversely, given a grading we can define theC ∗ action accord- } 683 ingly. �

684 Example 5.11.C[x,y], which corresponds to P 1 ; O C 685 Example 5.12. IfM= M is an object, thenM(m) is defined ⊕ n n 686 by the shifted grading,M(m) n =M n m − 687 Example 5.13. The Serre twisting sheaves are a particular case of 688 the last two examples. We have P 1 (i)=C[x,y](i), O C 689 Definition 5.14. We define the global sections functor for a graded 690 C[x,y]-module to just be the zeroeth graded component. Γ( nMn) = ⊕ 691 M0. Clearly, this coincides with the usual definition of global sections 692 of an 1 -module. O P 693 6. The Borel-Weil Theorem For an algebraic groupG, we say thatV is an algebraic module if we have a map to GL(V ) that is a morphism of group varieties. Given an algebraicB-moduleV , we can obtain another algebraicB-module 6. THE BOREL-WEIL THEOREM 29

(SL(2)) V by taking the right action ofB on (SL(2)). This space alsoO has a⊗ left (SL(2))-module structure. So, weO can define an induced (SL(2))-moduleO O IndSL(2)(V)=( (SL(2)) V) B B O ⊗ C 694 where the superscriptB denotes that we take theB invariant part 695 (only the vectors fixed byB via the action onV and the right action 696 on (SL(2))). OWe analyze how this induction works in more detail. Since we are considering SL(2), we will only need to work with one-dimensional algebraicB-modules, which we now characterize. A one-dimensional representation ofC ∗ ∼= Gm is a morphismC ∗ C ∗ respecting multi- plication, and it’s easy to see that these are the→ mapsz z n. There �→ are no non-trivial algebraic representations ofC ∼= Ga. Thus, the one- dimensional representations ofB are indexed by the integers. We let Cn denote the representation

a b n 1 1n =a − 1n 0a − � � 697 We have the following important result.

698 Theorem 6.1. (Borel-Weil)

SL(2) IndB Cn =V(n) ∗ B Proof. Consider the invariants ( (SL(2)) C n) . Note that the B-invariant submodules correspond exactlyO to⊗ irreducible submodules V (0), and hence to highest weight vectors of weight 0. We can use the Peter-Weyl theorem to write

B ∞ B ( (SL(2)) C n) = V(j) ∗ V(j) C n O ⊗ ⊗ ⊗ � j=0 � � NoteB only acts on the rightmost two factors, so we can reduce to

∞ B V(j) ∗ (V(j) C n) ⊗ ⊗ j=0 � 699 Now, for example, if v , . . . , v forms a basis forV , then v { 0 j} j { 0 ⊗ 700 1n, . . . , vj 1 n is a basis forV(j) C n. The only vector killed byE is ⊗ } ⊗ 701 vo 1 n, and it has weightj n. Thus, the only highest weight vectors ⊗ − SL(2) 702 of weight 0 occur whenj=n. So, we find Ind B Cn =V(n) ∗.� 7. BEILINSON-BERNSTEIN LOCALIZATION 30

703 Remark 6.2. More generally the Borel-Weil theorem implies that 704 forG semi-simple,B its Borel sub-algebra, every finite dimensional 705 representation ofG can be realized by induction fromB in this way.

706 What is the geometric interpretation of this theorem? We can re- 707 late the induced representation to line bundle structures on the quo- 708 tient SL(2)/B. By proposition ??, a one dimensionalB-moduleM 709 determines aG-equivariant (G/B) line bundle M˜. The global sec- O 710 tions Γ(M˜) of this line bundle have aG-action, and this module is G 711 IndBM. Let’s take a look at our example. We can describe quasi- 1 712 coherent - modules onP = SL(2)/B by consideringB-equivariant O ∼ 713 (SL(2))-modules. Starting from aB-moduleV , we can obtain such O 714 equivariant modules by tensoring (SL(2)) V and taking the right O ⊗ C 715 B-action on (SL(2)) as above. For example, starting withC n, our O 716 equivariant module will be (SL(2)) C n. By Borel-Weil the global O ⊗ 717 sections of the quotient bundle will beV(n) ∗, so we can identify this 718 line bundle with the twisting sheaf 1 (n). O P 719 7. Beilinson-Bernstein Localization 1 720 7.1.D-modules onP . In this section, we will construct certain 721 D-modules, which are essentially sets of solutions of algebraic differen- 722 tial equations. In section??, we will defineD-modules for any affine 2 2 2 723 algebraic variety, but for now, we consider the cases ofA ,A =A 0 1 ◦ \{ } 724 andP . To considerD-modules on a general algebraic variety, one sim- 725 ply sheafifies the construction for affine algebraic varieties.

726 Definition 7.1. We define the second Weyl algebra, W, to be the 727 algebra generated overC by x, y, ∂ x, ∂y , subject to relations [x, ∂x] = { } 728 [y,∂y] = 1, with all other pairs of generators commuting. W is a graded 729 algebra overC with degx = degy = 1, deg∂ x = deg∂ y = 1. − 2 730 Definition 7.2.AD-module onA is a module overW

731 Definition 7.3.AW -moduleM is torsion if for allm M, there k k ∈ 732 is ak such thatx m=y m=0

733 A similar consideration to that which led to quasi-coherent sheaves 2 734 onA yields the following ◦ 2 735 Definition 7.4. The category ofD-modules onA is the quotient ◦ 736 of the category ofW -modules by the full subcategory of torsion mod- 737 ules.

738 W contains a distinguished element, called the Euler operatorT= 2 739 x∂x +y∂ y. Geometrically,T corresponds to the vector field onA 7. BEILINSON-BERNSTEIN LOCALIZATION 31

740 pointing in the radial direction at every point, and vanishing only at 1 741 the origin. We now useW to defineD-modules onP : 1 742 Definition 7.5. The category ofD-modules onP has as its ob- 743 jects gradedW 2-modulesM modulo torsion such thatT acts on the 744 nth graded componentM n as scalar multiplication byn.

745 Remark 7.6. This graded action by the Euler operator is the cor- 746 rect notion of equivariance in the differential setting.

747 Example 7.7. The polynomial ringC[x,y] with the usual grading 1 748 is aD-module onP , wherex andy act by left multiplication, and 749 ∂x and∂ y act by differntiation. More generally, the structure sheaf is 750 always aD-module.

751 Example 7.8. The shifted modulesC[x, y](n) are notD-modules, 752 because although they are modules overW , the Euler operator does 753 not act on the graded components by the correct scalar.

1 754 Example 7.9.C[x,x − , y] with grading degx = degy = 1and 1 755 degx − = 1 is aD-module. Note that the global sections functor − 1 1 756 yields Γ(C[x, x− , y]) =C[x − y], whereas above we had Γ(C[x,y]) =C.

757 7.2. The Localization Theorem. We wish to investigate the 758 structure ofW a little further. If we decompose it into graded compo- 759 nents asW= i Z Wi, then what is the 0th componentW 0? Since ∈ 760 W acts faithfully onC[x, y], it suffices to consider the embedding � 761 W � End(C[x,y]) and answer the same question for the image of → 762 W.

763 Exercise 7.10. The componentW 0 is generated by the elements 764 x∂y,y∂ x, x∂x, andy∂ y. i j k l 765 Lemma 7.11. The elementsx y ∂x∂y form a basis forW 2.

766 Proof. Using the commutation relations, it is easy to show that 767 these elements are stable under left multiplication by the generators 768 ofW . Furthermore, since 1 is of this form, these elements must span 769 W . Thus it remains only to check the linear independence of these 770 elements. This is clear from the faithful action onC[x, y], so we are 771 done. �

772 Modifying the generating set forW slightly to bex∂ , y∂ , T, x∂ 0 y x x − 773 y∂y, we now notice a few interesting relations: x∂ (x)=0, y∂ (x)=y,(x∂ y∂ )(x) =x y x x − y x∂ (y)=x, y∂ (y)=0,(x∂ y∂ )(y)= y. y x x − y − 7. BEILINSON-BERNSTEIN LOCALIZATION 32

774 This is exactly the action of sl(2,C), where we identify the generators 775 E= x∂ ,F=y∂ , andH=x∂ y∂ , together with the elementT. y x y − x 776 Definition 7.12. LetU be a Hopf algebra acting on a module 777 A. ThenA is called a module algebra if we have a multiplicationµ: 778 A A A, which is a map ofU-modules. Specifically, if Δu=u u , ⊗ → 1 ⊗ 2 779 then we requireu(ab) = (u 1a)(u2b). 780 For the universal enveloping algebraU= (sl(2,C)) andx U ∈ 781 sl(2,C), we have the comultiplication map Δx=x 1+1 x, so the def- ⊗ ⊗ 782 inition of a module algebra imposes the conditionx(ab) = (xa)b+a(xb). 783 This is precisely the Leibniz rule, sox acts as a derivation. In particu- 784 lar, if (sl(2,C)) acts onC[x, y] as a module algebra, then the genera- U 785 torsE,F,H act as derivations and so their action coincides with that 786 of x∂ , y∂ , x∂ y∂ . (We leave it as an exercise to check thatU acts y x x − y 787 in the correct way.) 788 In particular, the action ofC x∂ y, y∂x, x∂x y∂y W End(C[x, y]) � − � ⊂ ⊂ 789 is identical to that of (sl(2,C)). Furthermore,T= x∂ x +y∂y is central U 790 insideW 0 since it acts as a scalar on each graded component and thus 791 commutes with these degree-preserving generators there. But we know 792 that the center of (sl(2,C)) is generated by the Casimir elementC, U 793 so we can expressT as a polynomial inC. SinceC acts onC[x, y] i as 794 scalar multiplication byi(i + 2), andT acts on it as multiplication by 2 795 i, we must haveC=T + 2T . Therefore we have 2 W0 = (sl(2,C))[T]/ C=T + 2T . U � � 1 796 For anyD-moduleM onP , we get an action ofW 0 on the global 797 sections Γ(M)=M 0. SinceT acts as zero onM 0, however, we see 798 that Γ(M) is in fact a module over (sl(2,C))/ C=0 . This is still U � � 799 an algebra, sinceC is central and thus C is a bi-ideal; we will let � � 800 0 = (sl(2,C))/ C=0 for convenience. U U � � 1 1 801 Example 7.13. IfM=C[x, x − , y] then Γ(M) =C(x − y), and 802 clearlyC acts on this by 0. We can compute the action ofE,F,H ∈ 803 sl(2,C) on this module (asx∂ y, y∂x, x∂x y∂ y respectively) to see that − 804 it is an infinite dimensionalsl(2,C)-module. Taking Fourier transforms 805 gives the dual of the Verma moduleM 0∗. 1 806 We now claim thatD-modules overP are equivalent to modules 807 overU 0. More precisely: 1 808 Proposition 7.14. The functorΓ:D-mod(P ) 0-mod is an →U 809 equivalence of categories.

810 Proof. Notice that Γ is representable by an objectD, i.e. Γ(M) ∼= 811 HomD-mod(D,M). (We leave it as an exercise to construct this object 7. BEILINSON-BERNSTEIN LOCALIZATION 33

1 812 D D-mod(P ) as a quotient ofW 2 by an elementT which is defined so ∈ 813 that the Casimir element acts the way it should, and to check that = U 0 814 End(D).) Thus we need to prove thatD is a projective. We require 815 two facts: first, that Γ = Hom (D, ) is exact, and second, that D-mod − 816 Γ is faithful, or that if Γ(M) = 0 thenM=0. 817 In order to prove exactness, we first need Kashiwara’s theorem: ifM 818 is torsion, thenM=C[∂ x, ∂y] M 0, whereM 0 = m M xm=ym= · { ∈ | 819 0 . We can check this for modules overW 1 =C x, ∂ x / [∂x, x] = 1 : } � � � � 820 for anyW -moduleM, we defineM = m M x∂ m= im . Then 1 i { ∈ | x } 821 we have well-defined mapsx:M i M i+1 and∂ x :M i M i 1, → → − 822 and x∂ :M M is an isomorphism fori< 0, so∂ x= x∂ + 1 x i → i x x 823 is an isomorphism onM fori< 1. But then both x∂ and∂ x are i − x x 824 isomorphisms onM , so in particularx:M M is an isomorphism i i → i+1 825 fori 2 and∂ x :M i M i 1 is an isomorphism fori 1. In ≤− → − ≤− 826 particular, if xm = 0, then x∂xm=(∂ xx 1)m= m and hence i − − 827 m M 1. More generally, ifx m = 0 then it follows by an easy ∈ − 1 828 induction thatm j−= i Mj. We conclude that ifM is torsion, then ∈ − 829 M=C[∂ x] M 1, and so the functorM M 1 gives an equivalence of · − � �→ − 830 categories from torsionW 1-modules to vector spaces. An argument by 831 induction will show that the analogous statement is true for anyW i, and 832 so in particular ifM is a torsionW 2-module thenM=C[∂ x, ∂y] M 2 · − 833 whereM 2 = m M Tm= 2m . Therefore any graded torsion − { ∈ | − } 834 W -module has all homogeneous elements in degrees 2. 2 ≤− 835 We can now prove that Γ is exact; since it’s already left exact, we 836 only need to show that it preserves surjectivity. Suppose that we have 837 an exact sequenceM N 0 in the category of graded modules → → 838 modulo torsion, so that in realityM N may not be surjective – → 839 all we know is thatC = coker(M N) is a graded torsion module. → 840 Taking global sections yields a sequence Γ(M) Γ(N) Γ(C), or → → 841 M N C , and sinceC is torsion we know that it is concentrated 0 → 0 → 0 842 in degrees 2, so thatC = 0. But Γ is exact in the graded category, ≤− 0 843 so the sequence Γ(M) Γ(N) 0 is exact as desired. Therefore Γ is → → 844 indeed exact. �

845 Exercise 7.15. Complete the proof by showing that Γ is faithful, 846 i.e. that ifM 0 = 0 thenM is torsion.

847 The representing objectDisa -module since End(D) = , so U 0 U 0 848 we now have a localization functor Loc(M)=D 0 M on the category ⊗ U 849 of -modules. This passes from an algebraic category to a geometric U 0 850 one, hence in the opposite direction from Γ. CHAPTER 4

851 The first quantum example:U q(sl2).

34 2. THE QUANTUM ENVELOPING ALGEBRAU q(sl2) 35

852 1. The quantum integers

853 In this section we introduce some polynomial expressions in a com- 854 plex variableq, called quantum integers, which share many basic arith- 855 metical properties with the integers. When we define the quantum 856 analogs ofSL 2 and sl2, the integral weights which arose there will be 857 replaced by quantum integral weights. The study of quantum integers 858 predates quantum physics, and goes back indeed to Gauss, who studied 859 q-series related to finite fields. Only in the last half of the twentieth 860 century have the connections between these polynomials and the math- 861 ematics of quantum physics come to be understood. The interested 862 reader should consult [?], [?], [?] for a more thorough exposition. Definition 1.1. Fora Z, we define the quantum integer, a a ∈ q q − a a 2 2 a a 1 [a]q = − =q +q − + q − +q − C[q,q − ]. q q 1 ··· ∈ − − 863 We will omit the “q” in the subscript when there is no risk of confusion.

864 We further define 865 (1) [a]!=[a][a 1] [1]. − ··· a [a]! 866 (2) = Z[q]. n [a n]![n]! ∈ � � − n qn 1 867 Exercise 1.2. Let (n)q :=q [n] 1 = − . Let q denote the field q 2 q 1 F − k 868 withq=p elements. Show that:

869 (1) The general linear group GLn(Fq) has order (n)q!. n n 870 (2) There are k q subspaces inF q of dimensionk. ¯ 1 1 (3) LetD, D:C�(q�)[x, x − ] C(q)[x, x − ] denote the difference operators, → 1 f(qx) f(q − x) f(qx) f(x) (Df)(x) := − ,( Df¯ )(x) := − . x(q q 1) x(q 1) − − − n n 1 n n 1 871 Show thatD(x ) = [n]x − , and D¯(x ) = (n)x − . Observe d 872 that lim D =lim D¯= . q 1 q 1 → → dx

873 2. The quantum enveloping algebraU q(sl2)

Definition 2.1. The quantum enveloping algebraU q(sl2) is the 1 1 C[q,q− ]-algebra with generatorsE,F,K,K − , with relations: 1 1 2 1 2 K K − − − − KEK =q E,KFK =q F,[E,F]= − 1 q q − 1 1 − KK− =K − K=1. 2. THE QUANTUM ENVELOPING ALGEBRAU q(sl2) 36

With these relations, we are equipped to prove the quantum analog 1 of the PBW theorem. DeclareE

877 Corollary 2.3.U q has no zero divisors.

878 Proof. This follows by computing the leading order coefficients in 879 the PBW basis. �

880 Remark 2.4. Observe that checking the diamond lemma forU q(sl2) 881 is actually slightly easier than for classicalsl 2. We will see that in many 882 ways the relations forU q(sl2) are easier to work with than for classical 883 U(sl 2).

884 We record the following commutation relations for future use:

m 1 1 m 1 m q − K q − K− m 1 885 Lemma 2.5. We have:[E,F ] = q−q 1 [m]F − . − − Exercise 2.6. Prove the lemma, using induction and the identity [a, bc] = [a, b]c+b[a, c].

An alternative proof of the PBW theorem for quantum sl2 may be given by constructing a faithful action ofU q, and verifying linear inde- pendence there. To this end, define an action ofU q on the vector space 1 A=C[x, y, z, z − ] as follows:

E(y sznxr) :=y s+1znxr,K(y sznxr) =q 2syszn+1xr, 1 s 1 s 1 s n r 2n s n r+1 s 1 zq − z − q − n r F(y z x ) =q y z x + [s]y − − z x . q q 1 − − 3. REPRESENTATION THEORY FORU q(sl2) 37

886 Exercise 2.7. Check that this defines an action, and verify that a b c a b c 887 E K F (1) =y z x . Conclude that the set of PBW monomials is 888 linearly independent.

n 889 In what follows, we will assume thatq = 1 for alln. The case where � 890 q is a root of unity is of considerable interest, and will be addressed in 891 later chapters . Notice that many of the proofs which follow depend 892 on this assumption. 893 Finally, we note in passing thatU q becomes a graded algebra if we 1 894 define deg(E)=1, deg(K) = deg(K − ) = 0,deg(F)= 1. −

895 3. Representation theory forU q(sl2)

896 The finite-dimensional representation theory forU q, whenq is not 897 a root of unity, is remarkably similar to that ofU, as we will see below. 898 Somewhat surprisingly, the representation theory ofU q whenqisa 899 root of unity is rather more akin to modular representation theory: 900 this arises from the simple observation that [m]q = 0 if, and only if, 2k 901 q = 1.

902 Definition 3.1. A vectorv V is a weight vector of weightλ if ∈ 903 Kv=λv. We denote byV λ the space of weight vectors of weightλ.A 904 weight vectorv V is highest weight if we also haveEv = 0. ∈ λ 905 Observe thatEV V 2 ,FV V 2 ; hence ifq is not a root of λ ⊂ q λ λ ⊂ q− λ 906 unity, andV is finite dimensional, we can always find a highest weight 907 vector.

Lemma 3.2. Letv V be a h.w.v. of weightλ. Definev 0 =v, [i] F i ∈ vi =F v0 = [i]! v. Then we have: i+1 1 i 1 2i λq− λ − q − Kvi =q − λvi, F vi = [i+1]v i+1, Evi = − vi 1. q q 1 − − − Proof. The first two are straightforward computations. For the third, we compute: i i 1 1 i 1 i 1 1 i i 1 1 EF q − K q − K− F − q − λ q − λ− Evi = v0 = − v0 = − vi 1. [i]! q q 1 [i 1]! q q 1 − − − − − − 908 �

909 Now, suppose V is finite dimensional andv 0 is a h.w.v. of weight 910 λ, andv m = 0, vm+1 = 0. Then, 0 = Evm+1 = [λ, m]v m, and thus m 1 m λq� − λ− q m m − 2 2m 911 [λ, m] = − 1 = 0. Hence λq− = λq , andλ =q λ= q q− −m − → 912 q . In conclusion, we have the following theorem. ± 4.U q IS A HOPF ALGEBRA 38

913 Theorem 3.3. For eachn 0, we have two finite dimensional ≥ n 914 irreducible representations of h.w. q of dimension n+1, and these ± 915 are all of the finite dimensional representations.

916 4.U q is a Hopf algebra

917 In this section we will see that the algebraU q is equipped with a 918 comultiplication and antipode making it into a Hopf algebra. These 919 will be modelled on the comultiplication and antipodes inU(sl 2) from 920 the previous chapter.

921 Proposition 4.1. There exists a unique homomorphism of algebras 922 :U U U defined on generators by � q → q ⊗ q 1 1 1 1 E=E 1+K E, F=F K − + 1 F, K ± =K ± K ± � ⊗ ⊗ � ⊗ ⊗ � ⊗ Proof. There is no problem defining Δ on the free algebraT= 1 C E,F,K,K− . In order for Δ to descend to a homomorphism from � � Uq(sl2), we need to check Δ(J)=0inU q(sl2) U q(sl2). For instance, we must check that: ⊗ 1 K K − (EF FE)=Δ( − ) � − q q q − − 923 This we will do now, and leave the remaining relations to the reader 924 to verify. 1 E F F E=(E 1+K E)(F K − + 1 F) � � −� � ⊗ ⊗ 1 ⊗ ⊗ (F K − + 1 F)(E 1+K E) − ⊗ 1 ⊗ ⊗ ⊗ 1 = (EF K − +E F+KF EK − +K EF) ⊗ 1 ⊗ ⊗ 1 ⊗ (FE K − +E F+FK K − E+K FE) − ⊗ ⊗1 ⊗ ⊗ = (EF FE) K − +K (EF FE) − 1 ⊗ ⊗ − 1 K K − 1 K K − − = − 1 K +K − 1 q q − ⊗ ⊗ q q − − 1 1 − K K K − K − = ⊗ − 1 ⊗ q q − − 1 K K − = ( − ) � q q 1 − − 925 �

926 Exercise 4.2. Verify that Δ is co-associative, and thus defines a 927 co-multiplication.

928 We can now define a co-unit� for Δ. Let�:U q C be the unique → 1 929 algebra map satisfying�(E)=�(F)=0,�(K)=�(K − ) = 1. 5. MORE REPRESENTATION THEORY FORU q 39

930 Exercise 4.3. Verify the co-unit axiom for� and Δ.

931 In conclusion, if we letµ andη be the multiplication and unit maps 932 on the algebraU q, we have that (Uq, µ, η,Δ,�) is a bi-algebra. We have 933 only now to produce an antipode. Proposition 4.4. There exists a unique anti-automorphismS of Uq defined on generators by: 1 1 1 S(K)=K − S(K − ) =KS(E)= K − ES(F)= FK. − − 2 1 934 Furthermore we haveS (u)=K − uK, for allu U . ∈ q Proof. There is no problem definingS on the free algebraT. To check thatS is well defined onU q then amounts to verifying that S(J) J, for which it suffices (sinceS is an anti-morphism) to check the statement⊂ on the multiplicative generators forJ. For instance, we must check: 1 K K − S(EF FE)=S( − ). − q q 1 − − 935 We do this now, and leave the remaining computations to the reader.

1 1 S(EF FE) =S(F)S(E) S(E)S(F)=FKK − E K − EFK − − 1 −1 K− K K K − =FE EF= − =S( − ) − q q 1 q q 1 − − − − 936 The remaining relations follow in similar spirit.�

937 5. More representation theory forU q

938 Now that we have equippedU q with the structure of a Hopf algebra, 939 its category of representations is endowed with a tensor product, as in 940 (??). In the classical case, we saw that the calculus of this tensor 941 product was rather simple, and could be expressed in terms of the 942 Clebsch-Gordan isomorphisms (??). In this section we will establish 943 the quantum Clebsch-Gordan isomorphisms, and we will show that the 944 categoryU q-mod is semi-simple. The formulations and proofs for both 945 statements will be completely analogous to the classical case.

946 Proposition 5.1.V (1) V (1) = V (2) V (0) + ⊗ + ∼ + ⊕ + Proof. We recall the notation of??:v 0 denotes a highest weight vector, whilev 1 =Fv 0. Consider the vectorv=v 0 v0 V +(1) V+(1). We have ⊗ ∈ ⊗ Ev=Ev v +Kv Ev = 0, Kv=Kv Kv =q 2v, 0 ⊗ 0 0 ⊗ 0 0 ⊗ 0 1 1 F v=Fv K − v +v Fv =q − v v +v v , 0 ⊗ 0 0 ⊗ 0 1 ⊗ 0 0 ⊗ 1 5. MORE REPRESENTATION THEORY FORU q 40

[2] [3] F v=v 1 v 1,F v=0. ⊗1 947 Finally, we have the vectorw=q − v v v v , such thatKw=w, 0 ⊗ 1 − 1 ⊗ 0 948 Ew=Fw = 0. �

949 These two submodules produce the required decomposition.

1 950 Definition 5.2. The character chV C[q,q − ] of a finite dimen- ∈ 951 sionalU (sl )-module is the trace ofK . q 2 | V 952 Exercise 5.3. Verify that chV(n) = [n + 1]q.

953 Exercise 5.4.[?] State and prove the general quantum Clebsch- 954 Gordan formula forU q(sl2), by mimicking our proof forU(sl 2).

955 Exercise 5.5. Letc :V (1) V (1) V (1) V (1) denote V (1),V (1) ⊗ → ⊗ 956 theU q(sl2)-linear endomorphism which scales the componentV (2) in 1 957 the tensor product byq, and the componentV (0) by q − . With − 958 respect to the basisv v , v v , v v , v v of the tensor product, 0 ⊗ 0 0 ⊗ 1 1 ⊗ 0 1 ⊗ 1 959 show that:

q 0 00 0q q 1 1 0 c = − . V (1),V (1)  0− 1 0 0   0 0 0q      960 5.1. Quantum Casimir element.

Definition 5.6. The quantum Casimir operator,C U q is the element ∈ Kq+K 1q 1 Kq 1 +K 1q C =FE+ − − =EF+ − − q (q q 1)2 (q q 1)2 − − − − 961 Exercise 5.7. Show that the two definitions ofC q are equal, and 962 thatC q is central.

963 Exercise 5.8. Let� +, , and letV �(m) be a simpleU q mod- ∈{ −} qm+1+q m 1 964 − − ule. ThenC q acts by the scalar�( q q 1 ). In particular,C q dis- − − 965 tinguishes between the differentV �(m)

q+q 1 966 ˜ − Thus, we have a central element Cq =C q (q q 1)2 , which acts as − − − 967 zero on a simple moduleM if and only if it is the trivial module.

968 Theorem 5.9. The category ofU q(sl2(C))-modules is semi-simple.

969 Proof. It is identical to the proof of the classical case??, using 970 Cq in place ofC. � 6. THE LOCALLY FINITE PART AND THE CENTER OFU q(sl2) 41

971 Remark 5.10. We have shown that whenq is not a root of unity, 972 the category of finite-dimensional type IU q-modules is equivalent to 973 the categoryU-mod, as abelian categories. However, as tensor cat- 974 egories, they cannot be equivalent, because the co-product is non- 975 cocommutative inU q.

976 Remark 5.11. For anyM a finite dimensionalU q-module, we can 977 decomposeM=M + M , whereM + is a sum of type I modules, ⊕ − 978 M is a sum of type II modules. Finally, we observe in passing that − 979 V (m) = V (0) V +(m). − ∼ − ⊗

980 6. The locally finite part and the center ofU q(sl2)

There is a peculiarity in the construction ofU q(sl2). As with any Hopf algebra, we may consider the “adjoint” action ofU q(sl2) on itself: x y :=x yS(x ), · (1) (2) 981 where Δ(x) =x x (the implicit sum is suppressed in the nota- (1) ⊗ (2) 982 tion). In the classical setting, the adjoint action is just the commutator 983 action, and we found (via the PBW theorem) thatU(sl 2) decomposed 984 naturally as a direct sum of finite dimensional representations. In par- 985 ticular, for any givenx U(sl ), the orbitU(sl ) x ofx was finite- ∈ 2 2 · 986 dimensional. For a Hopf algebraH, we letH � denote the sub-space of 987 elementsx which generate a finite orbit under the adjoint action. ForU q(sl2), we compute:

E (E lKmF n) · l+1 m n l m n 1 =E K F KE K F K− E − n 1 1 n 1 2l 2n+2m l+1 m n 2l 2n l m q − K q − K− n 1 = (1 q − )E K F +q − E K − [n]F − . − q q 1 − − F (E lKmF n) · =FE lKmF nK E lKmF nFK − n 1 1 1 n 2n 2m 2 l m+1 n+1 2n l 1 Kq − K − q − m+1 n =q (q− q )E K F q [l]E − − K F . − − q q 1 − −

It follows easily that the locally finite partU q�(sl2) ofU q(sl2) is generated 1 1 by the elementsEK − ,F,K− . Let us define:

2 1 1 1 K − E¯=EK − , F¯=F,K − , L¯= − . q q 1 − − 6. THE LOCALLY FINITE PART AND THE CENTER OFU q(sl2) 42

We can easily compute commutation relations amonst generators of U �(sl2): 2 ¯ ¯ ¯ ¯ 1 K − ¯ (3) EF F E= − 1 = L. − q q − 4−¯ 4 2 ¯ ¯ ¯ 2 4 ¯ ¯ ¯ ¯ q E q K− E E+ EK− 2 ¯ (4) q LE EL= − − 1 =q [2]E. − q q − 2 − 4 4 2 ¯ ¯ 4 ¯ ¯ F K − F q F q FK− 2 ¯ (5) LF q F L= − − 1− = q [2]F. − q q − − 1 2 − (6) (q q − )L¯=1 K − . − − ¯ ¯ 988 Proposition 6.1. The algebraU q�(sl2) is freely generated by E, F, 1 989 L¯, andK − , subject to relations (3)-(6).

Proposition 6.2. The specializationU 1� (sl2), ofU q�(sl2) atq=1, is isomorphic toU(sl 2) C[Z/2], via: ⊗ φ:U 1� (sl2) U(sl 2) C[Z/2], → ⊗ 1 (E,¯ F,¯ L,¯ K− ) (E,F,H,�), �→ 990 where� is the non-trivial element inZ/2. Proposition 6.3. We have an isomorphism,

k Uq�(sl2) = C[Z/2] Symq V (1) . ∼ ⊗ � k 0 � �≥ 991 Corollary 6.4. The center ofU q(sl2) is the subalgebra freely gen- 992 erated byC q.

993 Proof. The center of any Hopf algebra coincides with the ad- 994 invariant part, and so is clearly contained inU q�. The character com- 995 putation of Chapter 1 therefore applies mutatis mutandis.� CHAPTER 5

996 Categorical Commutativity in Braided Tensor 997 Categories

43 1. BRAIDED AND SYMMETRIC TENSOR CATEGORIES 44

998 1. Braided and Symmetric Tensor Categories

999 The Hopf algebras appearing in classical representation theory are 1000 either commutative as an algebra, or co-commutative as a co-algebra. 1001 Their quantum analogs are clearly no longer commutative, nor co- 1002 commutative; however they satisfy a weaker condition called “quasi- 1003 triangularity”, which we now explore. 1004 LetH be a Hopf algebra, and consider the tensor productV W of ⊗ 1005 H-modulesV andW . We have the mapτ:V W W V of vector ⊗ → ⊗ 1006 spaces, which simply switches the tensor factors,τ(v w)=w v. ⊗ ⊗ 1007 Exercise 1.1. Show thatτ is a morphism ofH-modules for all 1008 V,W H-mod if, and only if,H is either commutative or co-commutative. ∈ 1009 (hint: consider the left regular action ofH on itself)

1010 The tensor flipτ is not a map ofU q(sl2)-modules, asU q(sl2) is nei- 1011 ther commutative, nor co-commutative. Nevertheless, in this chapter, 1012 we construct natural isomorphismsσ :V W W V generalizing V,W ⊗ → ⊗ 1013 σV (1),V (1) from Chapter 4, and satisfying a rich set of axioms endowing 1014 the category =U (sl ) mod of finite dimensionalU (sl )-modules C q 2 − f q 2 1015 with the structure of a braided tensor category. Definition 1.2. Let( , , a, l, r) be a tensor category. A commu- tativity constraintσ on isC a⊗ natural isomorphism, C σ :V W W V, V,W ⊗ → ⊗ 1016 forV,W , such that for allU,V,W the following diagrams commute. ∈C σ U (V W) � (V W) U ⊗α ⊗ ⊗ ⊗� �� �� � � � (U V) W V (W U) ⊗ ⊗ � ⊗ � ⊗ �� �� σ �� �� σ (V U) W � V (U W) ⊗ ⊗ α ⊗ ⊗

(U V) W σ � W (U V) α⊗ ⊗ ⊗ ⊗ � α �� �� � � � U (V W) (W U) V) ⊗ ⊗ � ⊗ � ⊗ �� �� σ �� �� σ U (W V) � (U W) V ⊗ ⊗ α ⊗ ⊗ 1017 A braided tensor category is a tensor category, together with a 1018 commutativity constraintσ. 1. BRAIDED AND SYMMETRIC TENSOR CATEGORIES 45

1019 Remark 1.3. Like the pentagon axiom for the associator, the hexagon 1020 axiom resolves a potential ambiguity. To moveW pastU V , one can ⊗ 1021 either groupU andV, and commuteW past them jointly, or pass 1022 them one at a time. The hexagon diagram axiom asserts that in either 1023 manner, you obtain the same isomorphism.

1024 Remark 1.4. More concisely, a commutativity constraint is the op 1025 data necessary to equip the identity functor, Id : , from tothe op C→C opC 1026 category with the structure of a tensor functor, where denotes C op C 1027 the same underlying abelian category as , but withV W :=W V. C ⊗ ⊗ Proposition 1.5. Let be a braided tensor category, and letU,V,W . Then (suppressing associators),C we have the following equality in∈ HomC (U V W, W V U): C ⊗ ⊗ ⊗ ⊗ σ σ σ =σ σ σ . V,W ◦ U,W ◦ U,V U,V U,W V,W Proof. The naturality ofσ in each argument implies:

σV U,W σ V,W IdW =σ V,W σ U V,W . ⊗ ◦ ⊗ ◦ ⊗

1028 Applying the hexagon axiom to each ofσ V U,W andσ U V,W , we obtain ⊗ ⊗ 1029 asserted equality. �

1030 Definition 1.6. A braided tensor category is symmetric if for C 1031 eachV,W , we haveσ σ = Id. ∈C V,W ◦ W,V 1032 Exercise 1.7. LetH be a commutative or co-commutative Hopf 1033 algebra. Check thatσ=τ is a commutativity constraint onH-mod, 1034 and that it squares to the identity, so thatH-mod is a symmetric tensor 1035 category.

1036 Exercise 1.8. Denote byS n the symmetric group onn letters, 1037 generated by adjacent swapss . LetV be an element of a i,i+1 ∈C 1038 symmetric tensor category. Show that the maps i,i+1 Id. �→n ⊗···⊗ 1039 σ Id. defines a homomorphism ofS toEnd(V ⊗ ). In the case V,V ⊗· · ·⊗ n 1040 =H-mod, and dim V n, show that this is an inclusion. (Hint: C C ≥ 1041 consider a basise , . . . , e , and argue that the stabilizer ofe e 1 n 1 ⊗ · · · ⊗ n 1042 is trivial).

1043 Whenq is not a root of unity, we exhibited in Chapter ?? an 1044 equivalence of abelian categories between the category of finite dimen- 1045 sional type-IU q(sl2)-modules and that of the finite dimensionalU(sl 2)- 1046 modules. There we observed that as tensor categories these two are 1047 not equivalent, becauseU q(sl2) is non-cocommutative. 2. R-MATRIX PRELIMINARIES 46

1048 2. R-matrix Preliminaries In this section we answer the natural question: what is the necessary structure on a Hopf algebraH, to endow =H-mod with the structure of a braided tensor category? To answerC this question, let us suppose that the categoryH-mod is braided, and consider the left regular actoin ofH on itself. We have a braiding σ :H H H H. H,H ⊗ → ⊗ We defineR:=τσ H H (1 1). Given arbitraryH-modulesM andN, and arbitrary elements⊗ m⊗M,n N, we have a homomorphism of H-modules, ∈ ∈ µ :H H M N, m,n ⊗ → ⊗ h h h m h n. 1 ⊗ 2 �→ 1 ⊗ 2 1049 By naturality ofσ, we must haveσ (m n) =τR(m n). M,N ⊗ ⊗ 1050 Remark 2.1. For historical reasons relating to their physics origins, 1051 braiding operators are often calledR-matrices. ElementsR H H ∈ ⊗ 1052 obtained in this way are called “universal R-matrices”, as their action 1053 on anyV W is anR-matrix. ⊗ 1054 Exercise 2.2. Show that the elementR is invertible and satisfies op 1 op 1055 Δ (u)=RΔ(u)R − , where Δ =τ H,H Δ, or in Sweedler’s notation, op ◦ 1056 Δ (u)=u u . Hint: Apply theH-linearity ofc (2) ⊗ (1) V,W Exercise 2.3. Show that the hexagon relations imply the identity (Δ id)(R) =R 13R23 and (id Δ)(R) =R 13R12, where forR= ⊗s t , we define: ⊗ i ⊗ i �R13 := si 1 t i,R23 := 1 s i t i,R12 = si t i 1. ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 1057 Definition� 2.4. A quasi-triangular� Hopf algebra is� a Hopf algebra op 1058 H, equipped with an invertible elementR H H, such that Δ (u)= 1 ∈ ⊗ 1059 RΔ(u)R− for allu H, and satisfying (Δ id)(R) =R R and ∈ ⊗ 13 23 1060 (id Δ)(R)=R R . ⊗ 13 12 Exercise 2.5. LetH be a quasi-triangular Hopf algebra, withH- modulesM andN. DefineH-module homomorphisms, σ :M N N M, M,N ⊗ → ⊗ σ(m n) :=τ(R(m n)). ⊗ ⊗ 1061 Prove thatσ defines a braiding on the category ofH-modules.

1062 We have shown that the data of a braiding on the category ofH- 1063 modules is equivalent to that of a quasi-triangular structure onH. 4. LUSZTIG’SR-MATRICES 47

1064 3. Drinfeld’s UniversalR-matrix

1065 The first universalR-matrix forU q(sl2) was given by Drinfeld [?]. 1066 Drinfeld’s solution expresses the universalR-matrix forU q not as living 1067 inU q U q, but rather in a�-adic completion ofU q[[H,h]] Uq[[H,h]], ⊗ /2 ⊗ H 1068 � � whereH,h are formal parameters satisfyingq=e , andK = exp 2 : 2 n � ∞ n(n 1) (1 q ) − n n � R=( − q− 2 F E ) exp( H H) [n]! ⊗ 4 ⊗ n=0 � 1069 Passing to the formal completion here is unavoidable, if we wish to 1070 construct a universalR-matrix, and thus have naturalR-matrices on 1071 the entire category. However, if we restrict our attention only to the 1072 finite dimensional (or even the locally finite)U q-modules, we can ex- 1073 plicitly constructR-matrices there. As the discussion of completions 1074 would take us too far afield, we will not follow Drinfeld’s original ap- 1075 proach, but rather will follow Lusztig’s construction of the braiding on 1076 the category of locally finiteU q-modules.

1077 4. Lusztig’sR-matrices

To defineR-matrices on the category of type-I locally finiteU q- modules, it suffices to define them on the simple modulesV +(m), for m Z 0. In this section, we will do just this, following Lusztig’s construction∈ ≥ [?]. To begin, we define elements Θ inU U : n q ⊗ q n(n 1) 1 n n n n − (q q − ) Θ =a F E , a = ( 1) q− 2 − n n ⊗ n − [n]! 1 For example, Θ = 1 1,Θ = (q q − )F E. And we have 0 ⊗ 1 − − ⊗ 1 (n 1) q q − an = q − − − an 1. − [n] −

1078 Exercise 4.1. Prove the following identities: 1 (E 1)Θ n + (K E)Θ n 1 = Θn(E 1) + Θ n 1(K− E) − − ⊗ ⊗ 1 ⊗ ⊗ (1 F)Θ n + (F K − )Θn 1 = Θn(1 F)+Θ n 1(F K) ⊗ ⊗ − ⊗ − ⊗ (K K)Θ = Θ (K K) ⊗ n n ⊗ Exercise 4.2. Letα be an algebra (anti)automorphism of a Hopf algebraH, and define

1 α 1 α 1 α α S α − automorphism Δ = (α α) Δ α− , ε =ε α − ,S = ◦ ◦ 1 1 . ⊗ ◦ ◦ ◦ α S − α − antiautomorphism � ◦ ◦ 1079 Show that these define a Hopf algebra structure onH. 5. WEIGHTS OF TYPE I, AND BICHARACTERS (NEEDS BETTER TITLE) 48

1080 Exercise 4.3. There exists a unique antiautomorphismα:U q 1 → 1081 Uq such thatτ(E)=E,τ(F)=F,τ(K)=K − . Thus we can use this antiautomorphism to define an alternate Hopf algebra structure onU q, α 1 α α Δ (E)=E 1+K − E,Δ (F)=F K+1 F,Δ (K)=K K. ⊗ ⊗ ⊗ ⊗ ⊗ Definition 4.4. Define the linear operator Θ :M N M N by ⊗ → ⊗

Θ = Θn, n 0 �≥ 1082 where ˜r denotes multiplication on the� left byr.

1083 BecauseE,F act locally nilpotently on any locally finite module, 1084 this infinte sum is in fact a finite sum when applied to any vector, and 1085 thus is well-defined in End (M N). Because Θ = 1 1+ (locally C ⊗ ⊗ 1086 nilpotent operators) is unipotent, Θ is invertible.

Remark 4.5. For anyu U q, we have an equality of the linear maps ∈ Δ(�u)Θ=Θ Δ�α(u).

1087 If the righthand side were Δ�op(u) instead of Δ�α(u), then Θ would sat- 1088 isfy the same relations as a universalR-matrix??. This modification 1089 is accomplished in Drinfeld’s construction by the multiplying by the h 1090 infinite series exp( H H). However, as we will see, Lusztig’s solution 4 ⊗ 1091 still gives anR-matrix when restricted to the locally finiteU q-modules. Exercise 4.6. We compute the matrix Θ explicitly for the module V (1) V (1). Choose the standard basis forV (1) = span v 0, v1 and V (1)⊗V (1) = span v v , v v , v v , v v . Deduce:{ } ⊗ { 0 ⊗ 0 0 ⊗ 1 1 ⊗ 0 1 ⊗ 1} 1 0 0 0 0 1 0 0 Θ0 + Θ1 =  1  . 0q − q10 − 0 0 0 1     1092 5. Weights of Type I, and bicharacters (needs better title)

1093 Definition 5.1. A finite dimensional module forU q is type I if all n 1094 weight spaces are in Λ = q , n Z . { ∈ } 1095 Definition 5.2. We denote byχ(M) the character ofM, which is i 1096 the formal sumχ(M)= dimMqi z . We note that theχ(V(n))’s are 1097 linearly independent andM = N if, and only ifχ(M)=χ(N). � ∼ 5. WEIGHTS OF TYPE I, AND BICHARACTERS (NEEDS BETTER TITLE) 49

1098 Definition 5.3.A bi-character is a mapf:Λ Λ C × s.t. × → f(λλ �, µ) =f(λ, µ)f(λ �, µ),

f(λ, µµ �) =f(λ, µ)f(λ, µ �), f(λ, µ) = λf(λ, µq 2) = µf(λq 2, µ). Then we have

a b ab 2 1 1 f(q , q ) =f(q,q) , f(q,q)f(q,q)=f(q,q ) =q − f(q,1)=q − ,

1 1099 thusf(q,q) is a square root ofq − .

a b 1100 Exercise 5.4. Choose a square root ofq, and definef(q , q ) = ab 1101 q− 2 . Check that this gives a bi-character

For any finite dimensionalU -modM,M �, define f˜:M M � q ⊗ → M M � as follows, ⊗ form M , m� M � , f˜(m m �) =f(λ,µ)(m m �). ∈ λ ∈ µ ⊗ ⊗ f f f op 1102 Lemma 5.5. LetΘ = Θ f˜, then we haveΔ(u) Θ = Θ Δ (u). ◦ ◦ ◦ Proof. We need to check that Δα(u) f˜= f˜ Δ op(u) and it is enough to check for generators. For example,◦ we have◦

1 1 (E 1+K − E) f˜(m m �) =f(λ, µ)Em m � +λ − f(λ,µ)m Em � ⊗ ⊗ ◦ ⊗ ⊗ ⊗ 2 2 f˜ (E K+1 E)(m m �) =f(λq , µ)µEm m � +f(λ, µq )m Em �. ◦ ⊗ ⊗ ⊗ ⊗ ⊗ 1103 � f 1104 Theorem 5.6.Θ τ:M � M M M � is an isomorphism of ◦ ⊗ → ⊗ 1105 U -mod. Hereτ is the flip,τ(m n) =n m. q ⊗ ⊗ Theorem 5.7. The isomorphismsΘ f satisfy the hexagon relations ?? For all finite dimensional type IU q-modM,M �,M ��, the following diagrams commute:

M (M �� M �) (M M ��) M � ⊗ ⊗ −−−→ ⊗ ⊗ R R �  M (M� M ��)(M �� M) M � ⊗  ⊗ ⊗ � ⊗

 R � (M M�) M �� M �� (M M �) ⊗ � ⊗ −−−→ ⊗ ⊗ 6. THE YANG-BAXTER EQUATION 50

and (M � M) M �� M � (M M ��) ⊗ ⊗ −−−→ ⊗ ⊗

�  (M M�) M �� M � (M �� M) ⊗  ⊗ ⊗ � ⊗

 � M (M� M ��) (M � M ��) M ⊗ � ⊗ −−−→ ⊗  ⊗ 1106 Corollary 5.8.

1107 6. The Yang-Baxter Equation

1108 In this section, we show that the functorial isomorphismsR M,N : 1109 M N N M constructed above satisfy certain nice relations. → 1110 In particular, we letM 1,M2,M3 be three finite dimensionalU q mod- � � 1111 ules. We define operators onM 1 M2 M3 by lettingR 1,2 denote

1112 RM1,M2 1, and similarly forR 2,3; we defineR 1,3 as (1 P)(R 1,2)(1 P) 1113 whereP is the flip. Then the aim of� this� section is to prove � � � 1114 Theorem 6.1. With the above notation, we haveR 1,2R1,3R2,3 = 1115 R2,3R1,3R1,2.

1116 In order to prove this result, we will first introduce some auxiliary 1117 operators onM 1 M2 M3.

� n n n �� n 1118 Definition �6.2. Let� Θn =a nF K E and Θn =a nF n n ⊗ ⊗ ⊗ 1119 K− E . ⊗ Then we claim that these operators satisfy n �� (Δ 1)(Θ n) = (1 Θ n i)Θi ⊗ ⊗ − i=0 �n � (1 Δ)(Θ n) = (Θn i 1)Θi ⊗ − ⊗ i=0 � Proof. To prove this, we shall first describe the action of the co- product on powers ofE andF. In particular, we have r n i(r i) r i i i Δ(E ) = q − [r i]E − K E ⊗ i=0 �r n i(r i) i r i i Δ(F ) = q − [r i]F F − K− ⊗ i=0 � 6. THE YANG-BAXTER EQUATION 51

where we use [n i] to denote the quantum binomial coefficient. The proof of these identities goes by induction, the caser = 1 being the definition of the coproduct. To see the first identity in general: n n 1 Δ(E ) = Δ(E)Δ(E − ) r 1 − i(r 1 i) r 1 i i i = (E 1+K E)( q − − [(r 1) i]E − − K E ) ⊗ ⊗ − ⊗ i=0 r 1 � r − i(r 1 i) r i i i (i 1)(r 1) r i i 1 i = q − − [(r 1) i]E − K E + q − − [(r 1) (i 1)]KE − K − E − ⊗ − − ⊗ i=0 i=1 �r 1 �r − i(r 1 i) r i i i (i+1)(r 1) r i i i = q − − [(r 1) i]E − K E + q − [(r 1) (i 1)]E − K E − ⊗ − − ⊗ i=0 i=1 �r � i(r i) i r i r i i j = q − (q− [(r 1) i] +q − [(r 1) (i 1)])E − K E − − − ⊗ i=0 � i r i So we have reduced the question to showing (q− [(r 1) i] +q − [(r 1) (i 1)]) = [r i] which is an easy consequence of the− definition. The− formula− forF is proved by the same considerations. Now, we know n n that (Δ 1)(Θ n) =a n(Δ(F ) E ). Applying the above formula then yields ⊗ ⊗ n j(n j) j j n j n (q− − )[n j](a )F K − F − E n ⊗ ⊗ j=0 � n �� 1120 On the other side, computing i=0(1 Θ n i)Θi immediately yields n j n j j n ⊗ − n 2j(n j) j 1121 (an jaj)(F F − K− E ) which is equal to (an jaj)(q− − )(F j=0 − j=0 − j n j n ⊗ ⊗ � 2j(n j) ⊗ 1122 K− F − E ). So we see that it suffices to show (q− − )ajan j = � j(n j) ⊗ � − 1123 (q− − )[n j](an), but this is an easy computation from the definitions. 1124 The second formula follows from similar computations.� Now, the proof of the main theorem uses formulas obtained from the above via twisting byτ, whereτ is the antiautomorphism ofU q which 1 preservesE andF and sendsK toK − . Then we have (τ τ)(Θ n) = � �� ⊗ Θn, while (τ τ τ)(Θ n) = Θn. Applying (τ τ τ) to the above equations thus⊗ ⊗ yields ⊗ ⊗ n τ � ( Δ 1)(Θ n) = Θi(1 Θ n i) ⊗ ⊗ − i=0 �n τ �� (1 Δ)(Θn) = Θi (Θn i 1) ⊗ − ⊗ i=0 � 6. THE YANG-BAXTER EQUATION 52

We shall also need several more identities: if we define f˜ to be f˜ 1 1,2 ⊗ (and similarly for f˜2,3 and f˜1,3), then we have the relations f˜1,2Θ1,3 = � ˜ ˜ �� ˜ � � �� Θ f1,2 and f2,3Θ1,3 = Θ f2,3, where Θ = n Θn and similarly for Θ ; these relations follow immediately from the multiplicative properties of f˜. Further, one also easily derives �

f˜ f˜ (1 Θ) = (1 Θ) f˜ f˜ 1,2 2,3 ⊗ ⊗ 1,2 2,3 f˜ f˜ (Θ 1) = (Θ 1) f˜ f˜ 2,3 1,3 ⊗ ⊗ 2,3 1,3 To conclude that the Yang-Baxter equation holds, we write out both sides; the left hand being

(Θ 1) f˜ Θ f˜ (1 Θ) f˜ ⊗ 1,2 1,3 1,3 ⊗ 2,3 and the right hand being

(1 Θ) f˜ Θ f˜ (Θ 1) f˜ ⊗ 2,3 1,3 1,3 ⊗ 1,2 Now, using the above relations to rearrange the left hand side, one gets

(Θ 1)(Θ � )(1 Θ) f˜ f˜ f˜ ⊗ ⊗ 1,3 1,2 2,3 To deal with this, we rearrange the Θ terms as follows:

(Θ 1)(Θ � )(1 Θ) ⊗ ⊗ � = (Θ 1)(Θ i)(1 Θ n i) ⊗ ⊗ − n,i � = (Θ 1)( τ Δ 1)(Θ ) ⊗ ⊗ n n � = (Δ 1)(Θ )(Θ 1) ⊗ n ⊗ n � �� = (1 Θ n i)(Θi )(Θ 1) ⊗ − ⊗ n,i � = (1 Θ)(Θ �� )(Θ 1) ⊗ ⊗

1125 Where the third equality follows from the definition of Θ and the co- 1126 product. But this expression composed with f˜1,3f˜1,2f˜2,3 is precisely the 1127 right hand side; as is easily seen by using the above relations (and 1128 noting that the f˜’s all commute). 7. THE HEXAGON DIAGRAMS 53

1129 7. The hexagon Diagrams Theorem 7.1. The following diagrams commute: 1 R can R 1 M (M � M �� ) ⊗ M (M �� M � ) (M M �� ) M � ⊗ (M �� M) M � ⊗ ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ ⊗ = =

 can R can  M (M � M �� ) (M M � ) M �� M �� (M M � ) (M �� M) M � ⊗ � ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ � ⊗ and also R 1 can 1 R (M M � ) M �� ⊗ (M � M) M �� M � (M M �� ) ⊗ M � (M �� M) ⊗ ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ ⊗ = =

 can R can  (M M � ) M �� M (M � M �� ) (M � M �� ) M M � (M �� M) ⊗ � ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ ⊗ −−−→ ⊗ � ⊗ Proof. We shall prove the bottom diagram, the proof of the top is almost the same. In the top part of this diagram, the firstR is Θ1,2f˜1,2P1,2 while the secondRisΘ 2,3f˜2,3P2,3. Therefore, we consider the composition, which is

Θ2,3f˜2,3P2,3Θ1,2f˜1,2P1,2

= Θ2,3f˜2,3Θ1,3P2,3f˜1,2P1,2

= Θ2,3f˜2,3Θ1,3f˜1,3P2,3P1,2

�� = Θ2,3Θ f˜2,3f˜1,3P2,3P1,2

1130 where we have used the following equalities:P 2,3Θ1,2 = Θ1,3P2,3 and �� 1131 P2,3f˜1,2 = f˜1,3P2,3 and f˜2,3Θ1,3 = Θ f˜2,3. The last one was proved in 1132 the previous section, while the first two are immediate consequences of 1133 the definitions. Now, the lower half of the diagram involves only one 1134 R, which is given by the permutation (132) = (23)(12), followed by 1135 the diagonal matrix f˜(λµ, ν) (on a weight vector inM M M ), λ ⊗ µ ⊗ ν 1136 followed by (Δ 1)(Θ) (as the action on the tensor product is defined ⊗ 1137 by Δ). But we also have (Δ 1)(Θ) = (Θ )(Θ�� ), so combining this ⊗ 2,3 1138 with the relation f˜(λµ, ν)= f˜(λ,ν) f˜(µ, ν) shows that the two halves 1139 of the diagram are equal. � CHAPTER 6

1140 Geometric Representation Theory forSL q(2)

54 2. GENERATORS AND RELATIONS FORO q 55

1141 1. The Quantum Group Algebra

1142 In this section, we provide two independent constructions of a Hopf 1143 algebraO q(SL2), which plays the role of the coordinate algebraO(SL 2) 1144 in the quantum setting. First, we constructO q(SL2) as the algebra of 1145 matrix coefficients associated toU q(sl2), as in Chapter 2. Secondly, 1146 we introduce a simple non-commutative algebra called the quantum 1147 plane, and construct an algebra show thatO q(Mat2), the universal 1148 bi-algebra co-acting on the quantum plane, and inside there exhibit a 1149 central “q-determinant”, which we may set to one, to obtainO q(SL2).

1150 Definition 1.1. The quantized coordinate algebra,O q(SL(2)), 1151 henceforth denotedO q, is the subspace ofU q∗ spanned by matrix co- 1152 efficients of type I representations, i.e., linear functionals of the form 1153 c (u) :=f(uv), forV a type I representation ofU ,f V ∗, v V. f,v q ∈ ∈ ThatO q is a subalgebra follows immediately from the formula

cf,ecf �,e� =c f f �,e e� (as in the classical case). We giveO q a coalge- bra structure⊗ via⊗ Δ(c ) = c c . An antipode is obtained fi,ej k fi,ek ⊗ fk,ej from that onU q, via the bi-linear pairing betweenU q andO q: for a O , we defineS(a) by the� formula: ∈ q S(a),x = a, S(x) , � � � � 1154 wherex U is arbitrary. That this makesO into a Hopf algebra is ∈ q q 1155 an easy verification, just as in the classical case.

1156 2. Generators and Relations forO q

1157 Inspecting the proof of the Peter-Weyl Theorem for classicalSL 2, 1158 we see that the proof hinged only on the fact that the category of 1159 Uq(sl2)-modules is semi-simple, and that we had an explicit list of all 1160 simple objects. The category of finite-dimensionalU q(sl2)-modules is 1161 also semi-simple, and its simple objects are in bijection with those of 1162 U(sl 2). Thus we have:

Proposition 2.1. There exists an isomorphism ofU q(sl2) Uq(sl2)- modules, ⊗ Oq ∼= V ∗(k)�V(k). �k 1163 Remark 2.2. There is one subtlety in the construction ofO q: by 1164 design the algebraO(SL ) was equivariant forU(sl ) U(sl ). We 2 2 ⊗ 2 1165 should expect the same forO q, that it be equivariant for the action 1166 ofU U . However, there is a catch, which is that the antipode q ⊗ q 1167 S:U U is an anti-automorphism of the coproduct, i.e. Δ(S(x)) = q → q 2. GENERATORS AND RELATIONS FORO q 56

1168 S(x(2)) S(x (1)). This means thatO q is natural an algebra in the ⊗ op 1169 category , not . This is merely a reflection of contravariance C �C C�C 1170 of the duality functor : . ∗ C→C 1171 We wish to derive a “generators and relations” presentation ofO q, 1172 from its definition as matrix coefficients. To begin, we note that, as 1173 before, the moduleV (1) generates all finite dimensional representations n 1174 in the sense thatV(n) V (1) ⊗ (as follows from Clebsch-Gordan). ⊂ LettingC a, b, c, d denote the free algebra on symbols a, b, c, d, we have a surjection,� � C a, b, c, d O q, � �� (a, b, c, d) (c 0 , c 1 , c 0 , c 1 ). �→ f ,v0 f ,v0 f ,v1 f ,v1 Now, we can use theR-matrix to compute the commutativity relations between the matrix coefficients ofV (1), which we label a, b, c, d, where a=c 0,0,b=c 0,1,c=c 1,0 andd=c 1,1. We label theR-matrix entries k,l Ri,j and these are given by c (v v ) = Rk,lv v . V,V i ⊗ j i,j l ⊗ k Then we recall from the previous lecture� that we have 0,0 1,1 1 1,0 0,1 1,0 1 R =R =q − ,R =R = 1,R =q q − , 0,0 1,1 0,1 1,0 1,0 − 1175 and all remaining entries zero. These coefficients imply the following Lemma 2.3. The generators a, b, c, d satisfy the following relations: ab= qba, bc= cb, cd= qdc, ac= qca, 1 bd= qdb, ad da=(q q − )bc, ad qbc=1. − − − Proof. Each of the purely quadratic relations is obtained by ap- plying the relations, ij k l ij l k i j Rklaman = Rklcf f ,vm vn =c σ (f f ),vm vn ⊗ ⊗ ∗ ⊗ ⊗ ts j i ts � �i j i j =c f f ,σ(vm vn) = cf f ,vs vt Rmn = asatRmn, ⊗ ⊗ ⊗ ⊗ 1176 the so-called Fadeev-Reshetikhin-Takhtajian� (FRT) relations.� For instance,

0 0 0 0 qba=qc 0,1c0,0 =qc f f ,v1 v0 =c σ (f f ),v1 v0 ⊗ ⊗ ∗ ⊗ ⊗ 0 0 1 0 0 =c f f ,c(v v0) =c f f ,v0 v1 =c 0,0c0,1 = ab ⊗ ⊗ ⊗ ⊗ 1 0 0 1 ad=c 0,0c1,1 =c f f ,v0 v1 =c σ (f f ),v0 v1 ⊗ ⊗ ∗ ⊗ ⊗ 1 0 1 0 0 =c f f ,σ(v0 v1) =c f0 f1,v1 v0 + (q q − )cf f1,v v1 ⊗ ⊗ ⊗ ⊗ − ⊗ ⊗ 1 1 =c c + (q q − )c c =da+(q q − )cb. 1,1 0,0 − 1,0 0,1 − 3.O q COMODULES 57

1177 The remaining quadratic relations are proved similarly. The determi- 1178 nant relation follows, as in the classicalSL 2 computation.�

1179 Exercise 2.4. Using the PBW theorem forO q(SL2), show that 1180 the above generators and relations yield a presentation ofO q. (Hint: 1181 all the ingredients of Exercise ??) can be applied mutatis mutandis to 1182 the quantum setting.

1183 With this isomorphism in hand, we can explicitly exhibit the Hopf 1184 algebra structure onO . The comultiplication is given by Δ a, b, c, d = q { } 1185 a, b, c, d a, b, c, d where the right hand side symbolizes matrix { }⊗{ } 1186 multiplication; this is immediate from the above presentation ofO q. 1 1187 The antipode is given byS( a, b, c, d ) = d, qb, q − c, a ; one checks { } { − } 1188 easily that this is an antipode onSL q(2), and then we know from gen- 1189 eral Hopf algebra theory that if an antipode exists, it is unique; so it 1190 coincides with the antipode given by the pairing withU q.

1191 3.O q comodules

1192 LetM be a rightO q-comodule. Then we can put a leftU q-module 1193 structure onM as follows: by definition there is a map Δ :M → 1194 M O . Therefore we have mapsU M U M O M ⊗ q q ⊗ → q ⊗ ⊗ q → ⊗ 1195 U O M where the second to last map is the flip and the last q ⊗ q → 1196 is 1 <,>. By the properties of the Hopf pairing, this map makes ⊗ 1197 M into a leftU module. In particular, this association is a functor 1198 from rightO q comodules to leftU q modules, which, when restricted to 1199 finite dimensionalM, yields only type 1U q modules. This is because 1200 the weights ofK onM are given by coefficients of eigenvectors coming 1201 from expressions of the form foro O q; but the collection n ∈ 1202 of these is q n Z asO q is defined using only type 1 modules. Our { } ∈ 1203 remaining aim in this lecture is to show

1204 Theorem 3.1. The functor from finite dimensional rightO q co- 1205 modules to type1 finite dimensional leftU q modules is an equivalence 1206 of categories.

1207 Proof. In general, supposeC is a coalgebra andM a finite dimen- 1208 sional comodule. Let m , ..., m be a basis forM. Then we can write { 1 n} 1209 the coaction as Δm = m c . Now, coassociativity of this action i j j ⊗ j,i 1210 tells us that it is the same to apply ΔM 1 and 1 Δ C . The first gives � ⊗ ⊗ 1211 m c c , and so this implies that Δc = c c , or, j,k k ⊗ k,j ⊗ j,i i,j j k,j ⊗ j,i 1212 in matrix notation, Δ(cr,m) = (cr,m) (c r,m). Further, from the counit � ⊗ � 2 1213 axiom, �ci,j =δ i,j. Now, if you are given a collection ofn elements 1214 ofC called (c i,j), whose counit and comultiplication satisfy the above