Index

• (C (g,V), dg ); Chevalley–Eilenberg ρg, 285 complex of g, 286 σ-topology, 423 k C -regular Lie , 269 Diff( M), 287

Fω; flux homomorphism, 329 ∧κ, 298 M(k,σ1,...,σr); multi loop algebra, kω-space, 256 324 xl, left invariant vector field, 285 Pρ; flat bundle, 287 xr, right invariant vector field, 285 V (k), 283 A(P ); affine space of connections on P , W ∗-algebra, 423 321 [A, A], 78 V(M), 282 [ab], 76 (LB)-space, 260 Aut(P ), 282, 286 (LF)-space, 260 Bun(M,K), 328 2-morphism ΓE; sections of the bundle E, 286 betw. morph. of bun. ger., 342 Gau(P ), 282, 286 (P )=V(P )K , 282 ˙ aut AI (root system of type A), 62 V, 315 An (finite root system of type A), 63 z ⊕ω g; central extension defined by ω, abstract Wiener group, 399 294 abstract Wiener space, 397 gau(P ), 287, 314 accessible set, 425

Ad( P ), 316 adjoint action, 432 λg, 285 adjoint map, 421 1 Ω (M,V), 315 adjoint of a symmetric norming p Ω (M,V ), 294 function, 435 ω[p]; cocycles defined by a differential adjoint orbit, 432 form, 302 admissible Cartan , 173 ωη, 290 affine coadjoint action, 438, 443 ωκ, 290, 315 affine coadjoint orbit, 442 ωκ,η, 290 affine form, 74 Ω1(M,V ), 283 affine Kac–Moody Lie groups, 282 perκ, 297 affine Kac–Moody superalgebras, 183 perκ; κ invariant bilinear form, 297 affine Lie algebra, 65 perω; period homomorphism, 329 extended, 104 484 Index

locally extended, 108 automorphic forms on orthogonal toral type extended, 110 groups, 151, 157–160, 162–165, affine reflection Lie algebra, 96 167 isomorphic, 96 automorphic product, 157–160, nullity, 97 162–165, 167 affine reflection system, 72 classification, 164–165, 167 affine form, 74 reflective, 160, 163–165, 167 discrete, 75 morphism, 72 BI (root system of type B), 62 nullity, 72 Bn (finite root system of type B), 63 affine root system, 65 Banach ideal, 433, 434 extended, 75 Banach Poisson manifold, 420 locally extended, 75 Banach symplectic manifold, 420 Saito’s extended, 75 Banach–Lie algebra, 421 twisted, 67 Φ-reductive, 384 untwisted, 65 classical complex, 385 algebra, 76 classical real, 385 k-algebra, 76 Banach– base field extension, 79 Φ-reductive, 385 central, 77 Φ-reductive linear, 385 central-simple, 77 classical complex, 385 centre (associative), 77 classical real, 385 centre (Lie), 78 of Harish-Chandra type, 388 centroid, 76 Banach–Lie–Poisson space, 421 derivation, 76 base, 174 graded, see graded algebra, 77 invariant bilinear form, 76 BCI (root system of type BC), 62 perfect, 76 BCn (finite root system of type BC), 63 simple, 76 biconjugacy class, 359 almost local map, 262 bilinear form alternative torus, 23, 29 graded, 78 isotope, 29 invariant, 61, 76 (u ,u ) A 1 2 ,29 radical of, 61 isotopic, 30 Borel–Weil Theorem, 419, 444, 451 opposite, 31 Bott–Thurston cocycle, 309 amenable group, 375 bounded operator, 422 anisotropic roots, 100 boundedly regular, 249 Arens-type product, 377 box, 257 ARLA, see affine reflection Lie algebra, box topology, 257 96 Boyd indices, 371 associated norm ideals, 373 Brownian motion, 398 associated vector bundle, 445 Brownian sheet, 400 associative torus, 23, 24 bump function, 268 associative torus with , 23, 35 bundle ; Aut(P ), isotope, 37 282 (A, ι(h)), 37 bundle gerbe, 323, 341, 349, 354 isotopic, 39 basic, 347 mod-2 quadratic form, 36 bundle gerbe bimodule, 358 automatic smoothness, 275 bundle gerbe , 357 Index 485

CI (root system of type C), 62 coherent pre-reflection system, 60 Cn (finite root system of type C), 63 commutative space, 459 canonical map, 420 compact operator, 422 canonical projection, 72 compact regularity, 246 Cartan map, 290 compactly regular, 246 Cartan matrix, 174 compatible grading, 77 Cartan subalgebra complex analytic map, 252 splitting, 64 complex projective space, 451 Cartan–Helgason Theorem, 459 conditional expectation, 432, 445–447 Cartan–Helgason theorem, 464 cone, 252 CDer (centroidal derivations), 82 , 352, 358 Cent(A)(centroidofA), 76 connected component of Re(R), 60 (Cent A) ( centroidal transformations connected real part, 60 of degree ), 79 connected roots, 60 central, 77 connection central extension, 90, 339, 357, 435, on a bundle gerbe, 345 437, 442 on a line bundle, 344 covering, 90 consistent, 426 graded, 90 continuous vector bundle, 446 graded covering, 90 contractive bimodule, 371 central-simple, 77 contragredient Lie algebra, 169 centralizer, 432 contragredient Lie superalgebra, 170 centralizer of a functional, 430 core, 96 centralizer of an element, 430 coroot system, 62 centreless covering, 90 core, 96 crossed homomorphism of Lie algebras, Lie algebra, 77 304 centroid,9,76 crossed homomorphism of Lie groups, centroidal grading group, 81 306 centroidal derivation, 82 crossed product algebra, 80 centroidal grading group, 9, 81 curving, 345 Γ (L), 9 character, 194 DI (root system of type D), 62 character formula, 151, 153, 154 Dn (finite root system of type D), 63 characteristic distribution, 424, 426, D (degree derivations), 83 431, 434, 438–440, 442, 443 D-brane, 362 characteristic subspace, 424, 426, 429 Dahmen’s Theorem, 261 class 1 fundamental highest weight, 465 Darboux Theorem, 421 classical root systems, 62 ∆(E,H), 5 classification of locally finite root defect, 211 systems, 62 defining representation, 469 2-cocyle degree derivation, 83 group, 80 Deligne cohomology, 344, 346, 347, 355 invariant toral, 102 denominator identity, 151, 153, 154, coadjoint action, 428, 432, 438 160–163, 165, 166 coadjoint map, 421 Der derivation algebra, 76 coadjoint orbit, 350, 427, 431, 432, 438, (Der A) (derivations of degree ), 79 440, 443 derivation coefficient function, 467 centroidal derivation, 82 486 Index

degree derivation, 83 isotopy, 14–22 diffeomorphism group, 243 isotropic root, 6 direct limit, 252, 459 locally, 108 direct limit chart, 246 nullity, 6 direct limit properties, 245 toral type, 110 direct limit property, 268 type, 6 direct limit representation, 461 extended affine root system, 75 direct limit topology, 254 locally, 75 direct sum of pre-reflection systems, 60 Saito’s, 75 direct system, 252 extension discrete centralextension,90 extended affine Lie algebra, 107 graded, 90 reflection system, 75 datum, 69 divisible root, 62 of locally finite root system, 73 division-(R, Λ)-graded Lie algebra, 88 locally finite root system, 72 division-R-graded Lie algebra, 89 canonical projection, 72 division-graded associative algebra, 79 of pre-reflection systems, 69 division-root-graded Lie algebra, 88 universal central extension, 90 Dixmier–Douady class, 340 extension datum, 69 dual reductive pair, 461 of a locally finite root system, 73 duality pairing, 423, 433, 435–437, 441 extremely amenable group, 380

Eij (matrix unit), 112 faithful, 448 EALA, see extended affine Lie algebra, faithful functional, 430, 432 see extended affine Lie algebra fake monster algebra, 151, 160–163, 166 EARS, see extended affine root system fgc, 77 endomorphism fiber integral, 310 of degree ,78 finite growth, 173 equivalent crossed homomorphisms, 306 finite root system, 62 equivariant structure finite-dimensional superalgebras, 182 on a bundle gerbe, 353, 360 finite-rank operator, 434 Euler element, 225 flag geometries, 235 ev (evaluation map), 83 flux homomorphism, 329 even reflection, 179 formal degree, 461 evolution map, 247 formal degree renormalization, 466 exact invariant bilinear map, 290 Fq (the quantum torus associated to q), exponential function, 247 85 exponential function (of a Lie group), fragmentation map, 258 286 function algebra, 470 exponential map, 247 Fundamental Formula, 227 extended affine Lie algebra, 5, 104 fundamental highest weight, 464 centreless core, 6 construction, 10–13 Γ (centroidal grading group), 81 core, 6 gl (general linear Lie algebra), 112

discrete, 107 gl (general Lie algebra of row- and E(L, D, τ), 12 column-finite matrices), 119 family, 13 gauge group Gau(P ), 282 bijectively isomorphic, 22 Gelfand pair, 459 isomorphic, 6, 105 generalized foliation, 424 Index 487 generalized Kac–Moody algebra, Grassmann manifold, 476 151–156, 160–166 Grassmannian, 233 classification, 164–165, 167 group algebra, 77 construction, 160–161, 165–167 twisted, 80 generalized cohomology, 353 Lie algebra, 109 group manifold, 465 root system, 75 group of germs of diffeomorphisms, 244 geometric representation theory, 444 GRRS, see generalized reductive root GIF (graded invariant bilinear forms), system, 75 78 GNS representation, 444, 445, 447, 450 HC1(C) (first cyclic homology group of grCDer (graded centroidal derivations), C), 92 82 Hamilton equations, 420 grCent (graded centroid), 79 Hamiltonian vector field, 420–423, 427, grDer (graded derivation algebra), 79 437 grEnd (graded endomorphism algebra), Harish-Chandra decompositions, 388 79 Heisenberg group, 320 grSCDer (graded skew centroidal Hermitian vector bundle, 446 derivations), 82 highest vector, 195 graded algebra, 77 highest weight, 195 compatible grading, 77 highest weight module, 131, 132, 138, crossed product algebra, 80 141, 142 division-graded (associative), 79 highest weight representation, 462 division-graded (Lie), 89 highest weight vector, 462 full support, 78 Hilbert–Schmidt ideal, 370, 434 graded-central, 79 Hilbert–Schmidt operator, 422, 436 graded-central-simple, 79 holomorphic Hermitian vector bundle, graded-isomorphic, 78, 89 442 graded-simple, 78 holomorphic reproducing , 447 twisted, 80 holonomy homogeneous space, 77 of a bundle gerbe, 348 isograded-isomorphic, 78, 89 homogenous space, 77 Lie torus, 89 homomorphism, 427 multiloop, 81 homotopy group, 246, 273 predivision-graded (associative), 79 predivision-graded (Lie), 89 IDer (inner derivations), 84

quantum torus, 85 IDer (completed inner derivations), 119 root-graded, 88, 89 IARA, see invariant affine reflection division, 88 algebra, 100 invariant, 89 imaginary root, 59, 181 predivision, 88, 89 immersion, 425, 429 support, 77 indecomposable module, 128, 135, 139, torus, 79 143 Lie, 89 indecomposable pre-reflection system, graded-central algebra, 79 60 graded-central-simple algebra, 79 indivisible root, 62 graded-isomorphic, 5, 78, 89 infinite dimensional Lie group, 285 graded-simple, 78 inflation map, 331 grading subalgebra, 94 initial Lie , 274 488 Index inner ideal, 238 M algebras, 169 integrable module, 194 Kantor pair, 226 integrable root, 64 k[Λ] (group algebra of Λ), 77 integrable weight, 196, 209 kt[Λ] (twisted group algebra of Λ), 80 integral Kr¨amer classification, 468 diagonal, 372 k(S), 65 strictly lower triangular, 372 strictly upper triangular, 372 Lα (α-root space of L), 64 integral pre-reflection system, 60 Lagrangian geometry, 235 intrinsic subspace, 237 LB-space, 258 invariant leaf, 426 affine reflection algebra, 100 leaf of the characteristic distribution, isomorphic, 101 426 bilinear form, 61, 76 LEALA, see locally extended affine Lie root-graded Lie algebra, 89 algebra, 108 toral 2-cocycle, 102 LEARS, see locally extended affine root invariant 2-cocycle, 11 system, 75 invariant differential operator, 460 Leibniz property, 419 inverse of an invertible element, 88 LF-space, 258 invertible, 227 Lie algebra invertible element, 88 affine, 65 irreducible, 62 affine reflection, 96 isograded-isomorphic, 5, 78, 89 discrete extended affine, 107 ηgr,5 extended affine, 104 isomorphic generalized reductive, 109 affine reflection Lie algebras, 96 locally extended affine, 108 extended affine Lie algebras, 105 root-graded, 89 invariant reflection algebras, 101 simply connected, 90 isotope, 90 Steinberg, 116 isotopic, 90 toral type extended, 110 Isotrivilaity, 46 Lie algebra (filtered), 236 isotropic root, 173 Lie algebra (graded), 224 iterated loop algebra, 324 Lie algebra (symmetric), 224 Lie group Jandl structure, 360 compact, 357 Jordan algebra, 227 simply-connected, 349, 357 Jordan pair, 225 Lie subgroup, 428 Jordan torus, 23, 25 Lie torus, 7 isotope, 25 bi-isograded-isomorphic, 8 (u) A ,25 ϕr and ϕe,8 isotopic, 25 bi-isomorphic, 8 Jordan triple system, 226 centreless, 8 Jordan-H¨older series, 134 coordinatization theorems, 22–24 Jordan-Lie functor, 226, 233 grading pair, 8 isotope, 13 k-algebra, 76 L(s),13 K¨ahler form, 442 isotopic, 14 K¨ahler manifold, 451 multiloop, 41 Kac–Moody Lie superalgebra, 170, 179 root grading and external grading, 7 Index 489

type, 7 mononormalizing, 370 untwisted, 9 moonshine for Conway’s group, 160–163 Lietriplesystem,224 morphism, 422 Lie-(R, Λ)-torus, 89 between bundle gerbes, 342, 347 Lie–Poisson bracket, 422 of affine reflection systems, 72 limit aligned representation, 461 multiloop algebra, 49, 81, 324 limit irreducible representation, 458 multiplicity, 201 line bundle, 340, 345 multiplicity-free, 459 over a coadjoint orbit, 350 over the loop space, 355 natural representation, 127, 128, 130, linear Poisson map, 422 131, 137, 138, 141, 144, 145 Littlewood-Richardson coefficients, 129 negative root, 172 local map, 262 nest algebra, 373 locally kω space, 256 non-isotropic root, 173 locally convex direct limit, 248 non-regular Lie group, 272 locally convex direct limit topology, 258 non-symmetrizable superalgebra, 186, locally exponential Lie group, 275, 286 193 locally extended nondegenerate pre-reflection system, 60 affine Lie algebra, 108 nontrivial Boyd indices, 371 affine root system, 75 norm ideal, 369 locally finite Lie algebra, 127, 144, 264 normal, 432, 441, 448 locally finite root system, 62 normal functional, 432 classical, 62 normalized form, 63 classification, 62 normalized matrices, 172 coroot system, 62 null roots, 100 divisible root, 62 null-system, 234 extension datum, 73 nullity indivisible root, 62 of an affine reflection Lie algebra, 97 irreducible, 62 of an affine reflection system, 72 normalized form, 63 root basis, 62 octonion torus, 29 Loewy length, 129, 133, 139, 142 odd reflection, 176 logarithmic derivative, (left/right) orbit symplectic form, 429 δl/r(f), 286 long root, 65 psl (projective special linear Lie loop algebra, 324 algebra), 114 multiloop 81 pair geometry (linear, affine), 231 twisted, 67 parabolic direct limit, 472 untwisted, 66 parabolic direct system, 472 , 281 partialsection,68 loop space, 355 partition, 129 perfect algebra, 76 Mat (finitary matrices), 112 period homomorphism, 329

Mat (row- and column-finite matrices), pointed reflection subspace, 70 119 Poisson bracket, 421 Mackey complete, 247 Poisson Lie group, 440 mean, 376 Poisson manifold, 419 left invariant, 375 Poisson map, 420 Meyberg’s Theorem, 227 Poisson-Lie group, 437, 438 490 Index polar decomposition, 448 graded-isomorphic, 89 positive definite reproducing kernel, 446 isograded-isomorphic, 89 positive functional, 432 R-graded Lie algebra, 89 positive root, 172 (pre)division, 89 pre-reflection system, 59 Ran (anisotropic roots), 100 affine form, 74 R char (coroot system of R), 62 coherent, 60 R div (divisible roots of R), 62 direct sum, 60 R im (imaginary roots of R), 59

extension, 69 Rind (indivisible roots of R), 62 indecomposable, 60 R Ê (integrable roots of R), 64 integral, 60 Rre (real roots of R), 59 invariant bilinear form, 61 R× (the non-zero roots of R), 88 × morphism, 60 Rind (the non-zero indivisible roots of partial section, 68 R), 88 nondegenerate, 60 R0 (null roots), 100 quotient, 69 Rad (radical of a bilinear form), 61 real part of, 60 Re(R) (the real part of R), 60 connected, 60 real analytic map, 252 connected component, 60 real part of a pre-reflection system, 60 reduced, 60 real root, 59, 181 root string, 61 realization operator, 447–449, 451 unbroken, 61 reduced pre-reflection system, 60 strictly invariant bilinear form, 61 reduced root system, 7 symmetric, 60 reductive Lie algebra, 369 tame, 60 reductive Lie group, 368 predivision-(R, Λ)-graded Lie algebra, reflection space, 70 88 reflection subspace, 70 predivision-R-graded Lie algebra, 89 pointed, 70 predivision-graded associative algebra, symmetric, 70 79 reflection system, 59 predivision-root-graded Lie algebra, 88, affine, 72 89 associated to bilinear forms, 63 predual, 423, 428, 430, 433, 436–438 reflective root, 59 principal root, 176 regular Cartan matrix, 173 product integral, 247 regular contragredient superalgebra, product map, 258 176 projective elementary Lie algebra, 28 regular Lie group, 247, 285

pe3(A), 28 regular root, 173 projective geometry (generalized), 230 regular weight, 205, 209 projective group (elementary), 230 reproducing kernel, 390, 446, 447 restricted Banach algebra, 436 quantum matrix, 85 restricted , 396 quantum torus, 24, 85 restricted Grassmannian, 441–443 quasi immersion, 425 restricted unitary algebra, 435, 436 quasisimple, 173 restricted , 436, 438, 439 quotient pre-reflection system, 69 restriction map, 331 quotient root system, 72 riemannian symmetric space, 462 (R, Λ)-graded Lie algebra (R, Λ)-graded Lie algebra, 88 isotope, 90 Index 491

isotopic, 90 set of weights, 194 root, 172 short root, 65 anisotropic, 100 Silva space, 260 divisible, 62 simple algebra, 76 divisible, 62 simple roots, 172 imaginary, 59 simply connected, 90 integrable, 64 singular root, 173 long, 65 small subgroup, 274 null, 100 small , 248 real, 59 smooth bump function, 268 reflective, 59 smooth generalized distribution, 426 short, 65 smooth vector bundle, 446 root basis, 62 smoothly regular space, 268 root space decomposition, 64 Sobolev–Lie group, 244 root string, 61 socle filtration, 128, 129 unbroken, 61 special linear Lie algebra, 32 root system, slr+1(A), 32 affine, 65 special symplectic Lie algebra, 34

twisted, 67 ssp2r(A, ι), 34 untwisted, 65 spherical function, 467 extended affine, 75 spin factor, 229 finite, 62 split Lie subgroup, 427 generalized reductive, 75 splitting Cartan subalgebra, 64 locally finite, 62 squeezed bundle, 314 quotient, 72 standard base, 175 root-graded Lie algebra, 88, 89 standard imbedding, 224 grading subalgebra, 94 state, 237, 431, 432, 444, 445, 447, 448 invariant, 89 Steinberg Lie algebra, 116 root-reductive Lie algebra, 144 Stinespring dilation, 390 Stone–Weierstrass theorem, 471 S lg (long roots), 65 strange twisted affine superalgebra, 185,

Ssh (short roots), 65 193 SS(β, α)(α-root sting through β), 61 strict direct system, 254 slI (A) (special linear Lie Lie algebra), strictly invariant bilinear form, 61 112 string, 178 st (Steinberg Lie algebra), 116 strong symplectic form, 420, 429, 442, sl2-triple, 64 443 sα (reflection in α), 59 strong symplectic leaf, 434, 438, 439 Saito’s extended affine root system, 75 strong symplectic manifold, 421, 431 Schatten ideal, 370, 434 strongly Ck-regular Lie group, 269 Schur functor, 129 subalgebra Schur , 466 ad-diagonalizable, 64 Schur-Weyl Duality, 133, 135, 136, 146 toral, 64 SEARS, see Saito’s extended affine root subordinated cohomology class, 302 system, 75 subsystem, 60 section, 446 support, 77 self-adjoint element, 430 full, 78 self-adjoint functional, 430 symmetric norming function, 369, 434, Serre’s relations, 180 435 492 Index symmetric pre-reflection system, 60 two-cocycle, 437 symmetric reflection subspace, 70 types I and II, 174 symmetric space, 232 typical weight, 200, 209 symplectic leaf, 425, 426, 429, 434 uce (universal central extension), 90 tr (trace), 112 unbroken root string, 61 tame, 217 uniformly continuous, tame pre-reflection system, 60 left, 375 tame toral pair, 96 right, 375 tensor algebra, 128 unitary group, 430, 436 tensor representation, 128, 130–132, unitary orbit, 430, 431, 433 135–141, 144 universal ambit, 379 test function group, 244 universal central extension, 90 tier number, 65 untwisted affine root system, 65 Tits-Kantor-Koecher Lie algebra, 24 untwisted loop algebra, 66 TKK(A), 24 Verma module, 195 with Lie algebra, 275 vertex algebra, 151, 153, 155, 157, 160, toral pair, 96 162, 165, 166 centreless core of, 96 , 156, 160, 162 core of, 96 vertex operator algebra, 155–157, 165, tame, 96 166 toral subalgebra, 64 WZW model, 156 toral type extended affine Lie algebra, Virasoro group, 282, 308 110 toroidal Lie algebra, 9, 314 W (R), 60 Torsor, 46 weak direct limit chart, 246 torus weak direct product, 244, 257 associative, 79 weak immersion, 425, 429–431, 434 Lie, 89 weak K¨ahler, 433–435 quantum, 85 weak symplectic form, 420, 429, 438 trace class operator, 422, 436 weak symplectic leaf, 435, 439, 443 tracial, 432, 448 weak symplectic manifold, 421, 431, 433 transversal, 230, 235, 238 weight module, 194 triangular decomposition, 172 Weil representation, 158, 159, 164 trivialization well-filled chart, 272 of a bundle gerbe, 347 Wengenroth’s Theorem, 265 truncation Wess–Zumino–Witten model, 361 diagonal, 371 Weyl group, 60, 179 strictly lower triangular, 372 Weyl groupoid, 176 strictly upper triangular, 372 twisted affine Kac–Moody Lie groups, Yamasaki’s Theorem, 257 282 Young projector, 129 twisted affine root system, 67 twisted affine superalgebras, 184 Z(β,α), 61 twisted loop algebra, 67 Z(.) (the centre of an algebra), 77, 78