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List of Frequently Used Symbols
List of Frequently Used Symbols
The reader should note that some symbols have different meanings depending on the context. The Notational Conventions (page li) should also be consulted. The numbers to the right are the pages on which the symbols are explained.
Chapter 1 d 8
[*?.I
End n /
Cent 8' X
x,>Q va
LI
derivation, 3 Lie algebra, 3 Lie bracket, 3 algebra of endomorphisms, 4 representation, 4 quotient, 6, 9 center, 7 commutator of 8. 7 direct product, 7 semidirect product, 8 subspace of V of degree a,8 direct sum, 8
493
Wst of Frequently Used Symbols
494
degree, 8 tensor product, 10 induced module, 11, 13, 15 group algebra of G, 12 automorphism group, 12 basis of F[G], 12 tensor algebra of V, 13 nth tensor power of V, 13 symmetric algebra of V, 14 nth symmetric power of V, 14 universal enveloping algebra of a, 14 symmetric bilinear form, 16, 17 algebra of Laurent polynomials in t, 17 d d=tdt
derivation of F[t, t-'1, 17 central element, 18 affine Lie algebra, 18 extended affine Lie algebra, 18 affine Lie algebra twisted by an involution B (resp., extended twisted algebra), 20 positive (resp., negative) subalgebra of l, 21 induced modules, 22, 26 finite-dimensional abelian Lie algebra, 24 untwisted affinization of b, 25 twisted affinization of b (twisted Heisenberg algebra), 25 grading set, 25 symmetric bilinear or Hermitian form, 26, 30 derivations of F[t, f'],31 Witt algebra, 31 Viragoro algebra, 32 generators of the Virasoro algebra, 32 representation of h 6 t" (resp., of L,,), 34. 36 distinguished element of T2(b),35 vacuum vector of ~(Iji), 40 weight, 41 graded dimension, 42 Dedekind q-function, 45
Chapter 2 Uzl, VIZI, V[Z,z-'I, V"Zll% m z , z-'11 d(Z)
certain spaces of formal series, 48, 49 formal &function, 50
List of Frequently Used Symbols
495
specialization, 51 derivation, 56 formal integration, 57 generating function, 58 8-symmetrization (resp., antisymmetrization), 60
involutions of et(2, IF), 63, 64, 66 twisted space, 67 exponential series, 69 twisted vertex operator, 70 normal ordering, 73 positive (resp., negative) part of generating function, 74 symmetrization (resp., antisymmetrization) of vertex operator, 80
exponential series, 85 group algebra element, multiplication operator, 86 untwisted vertex operator, 87 untwisted spaces, 87, 89 positive (resp., negative) part of generating function, 89 normal ordering, 90 isomorphism of twisted and untwisted space, 98
Chapter 5 (K) &O
CO
A
=A/U
T
group generated by K , 102 2-cocycle, 103 commutator map, 104 abelian 2-group, 11 1 induced module, 118
Chapter 6 L Lm
Lo 8, (4)
lattice, 122 set of elements of square length rn, 123 dual lattice, 123 theta function, 125
496
List of Frequently Used Symbols
central extension, 126 root system, 127 Lie algebra associated to A, 127 root vector, 127, 128 multiplicative 2-cocycle, 128 root lattice of 8, 133 weight lattice of g, 133 root lattices and Lie algebra, 134 lifting of -1, 138
Chapter 7 induced L-module, 146 homomorphism, 146 operator in F[Ll, 147 untwisted vertex operator, 148 affine Kac-Moody algebras, 156 e-modules, 158, 167 twisted space, 158 twisted vertex operator, 159 twisted affine Kac-Moody algebras, 169
Chapter 8 field of rational functions, 175 field of fractions of formal power series ring, 175 linear maps, 176 one-parameter group, 180 untwisted generating function (new definition), 191 nonnegative (resp., negative) part of generating function (new definition), 191 normal ordering (new definition), 191 untwisted vertex operator, 192 residue, 195 general untwisted vertex operator, 198 component of vertex opeator, 200 vector corresponding to Virasoro algebra, 217 products, 229, 230 components of products, 229 cross-bracket, 229 involution of V,, 239
List of Frequently Used Symbols
VL*B
f
a
rank V
f 1-eigenspaces of 8, 239 subspace of &* of weight 2, 240 commutative afinization, 241 rank of vertex operator algebra, 245
Chapter 9 twisted generating function (new definition), 257 positive (resp., negative) part of generating function (new definition), 257 normal ordering, 257 twisted vertex operator, 258 quadratic differential operator, 264 general twisted vertex operator, 266 involution of &,' 274 *l-eigenspace of 8, 274 subspace of ' V of weight 2, 295
Chapter 10 finite set, 298 binary linear code, 298 dual code, 299 Mathieu group, 302 lattices associated with e, 303 Leech lattice, 304 &-root lattice, 305 Niemeier lattices, 308 Conway groups, 3 10 canonical &module, 3 13 moonshine module, 3 15 sum of untwisted and twisted modules, 315 Griess algebra, 316 affinization of Griess algebra, 317 centralizer of involution in M, 324 central involution in C, 325 graded character, 330 modular group, 336 modular invariant with zero constant term, 337
497
498
List of Frequently Used Symbols
Chapter 11
Q
Lo, L , , L
v, KO, vi#l,v,, VL,, VLl,
8
00 0 1
root lattice of type A?, 343 lattices associated with C?,343 untwisted spaces, 344 twisted spaces, 345 Lie algebra of type At4, 348 involution of 8 , 349 involution of W,, 349, 350, 364, 371
Chapter 12 24-element set, 375 binary Golay code, 375 lattices associated with A, 376 cosets associated with A, 376, 389 groups associated with lattices, 380, 383 spaces associated with a union of cosets M of Q, 385 automorphism of W and @, 396 automorphism of V , 397
6
Monster, 400 Q-form of Vh, 408 M-invariant positive definite Hermitian form on Vh, 41 I
Chapter 13 N H Ij
A
certain group, 420 certain group, 423 group associated with N(A?), 431 group associated with N(D,6),441
Appendix P -End, Hom R
infinite product of graded spaces, 462 spaces of linear operators from sums to products, 463, 464 analytic continuation, 466
L 2 n i ,[ dz
contour integral, 473
The reader should note that some symbols have different meanings depending on the context. The Notational Conventions (page li) should also be consulted. The numbers to the right are the pages on which the symbols are explained.
Chapter 1 d 8
[*?.I
End n /
Cent 8' X
x,>Q va
LI
derivation, 3 Lie algebra, 3 Lie bracket, 3 algebra of endomorphisms, 4 representation, 4 quotient, 6, 9 center, 7 commutator of 8. 7 direct product, 7 semidirect product, 8 subspace of V of degree a,8 direct sum, 8
493
Wst of Frequently Used Symbols
494
degree, 8 tensor product, 10 induced module, 11, 13, 15 group algebra of G, 12 automorphism group, 12 basis of F[G], 12 tensor algebra of V, 13 nth tensor power of V, 13 symmetric algebra of V, 14 nth symmetric power of V, 14 universal enveloping algebra of a, 14 symmetric bilinear form, 16, 17 algebra of Laurent polynomials in t, 17 d d=tdt
derivation of F[t, t-'1, 17 central element, 18 affine Lie algebra, 18 extended affine Lie algebra, 18 affine Lie algebra twisted by an involution B (resp., extended twisted algebra), 20 positive (resp., negative) subalgebra of l, 21 induced modules, 22, 26 finite-dimensional abelian Lie algebra, 24 untwisted affinization of b, 25 twisted affinization of b (twisted Heisenberg algebra), 25 grading set, 25 symmetric bilinear or Hermitian form, 26, 30 derivations of F[t, f'],31 Witt algebra, 31 Viragoro algebra, 32 generators of the Virasoro algebra, 32 representation of h 6 t" (resp., of L,,), 34. 36 distinguished element of T2(b),35 vacuum vector of ~(Iji), 40 weight, 41 graded dimension, 42 Dedekind q-function, 45
Chapter 2 Uzl, VIZI, V[Z,z-'I, V"Zll% m z , z-'11 d(Z)
certain spaces of formal series, 48, 49 formal &function, 50
List of Frequently Used Symbols
495
specialization, 51 derivation, 56 formal integration, 57 generating function, 58 8-symmetrization (resp., antisymmetrization), 60
involutions of et(2, IF), 63, 64, 66 twisted space, 67 exponential series, 69 twisted vertex operator, 70 normal ordering, 73 positive (resp., negative) part of generating function, 74 symmetrization (resp., antisymmetrization) of vertex operator, 80
exponential series, 85 group algebra element, multiplication operator, 86 untwisted vertex operator, 87 untwisted spaces, 87, 89 positive (resp., negative) part of generating function, 89 normal ordering, 90 isomorphism of twisted and untwisted space, 98
Chapter 5 (K) &O
CO
A
=A/U
T
group generated by K , 102 2-cocycle, 103 commutator map, 104 abelian 2-group, 11 1 induced module, 118
Chapter 6 L Lm
Lo 8, (4)
lattice, 122 set of elements of square length rn, 123 dual lattice, 123 theta function, 125
496
List of Frequently Used Symbols
central extension, 126 root system, 127 Lie algebra associated to A, 127 root vector, 127, 128 multiplicative 2-cocycle, 128 root lattice of 8, 133 weight lattice of g, 133 root lattices and Lie algebra, 134 lifting of -1, 138
Chapter 7 induced L-module, 146 homomorphism, 146 operator in F[Ll, 147 untwisted vertex operator, 148 affine Kac-Moody algebras, 156 e-modules, 158, 167 twisted space, 158 twisted vertex operator, 159 twisted affine Kac-Moody algebras, 169
Chapter 8 field of rational functions, 175 field of fractions of formal power series ring, 175 linear maps, 176 one-parameter group, 180 untwisted generating function (new definition), 191 nonnegative (resp., negative) part of generating function (new definition), 191 normal ordering (new definition), 191 untwisted vertex operator, 192 residue, 195 general untwisted vertex operator, 198 component of vertex opeator, 200 vector corresponding to Virasoro algebra, 217 products, 229, 230 components of products, 229 cross-bracket, 229 involution of V,, 239
List of Frequently Used Symbols
VL*B
f
a
rank V
f 1-eigenspaces of 8, 239 subspace of &* of weight 2, 240 commutative afinization, 241 rank of vertex operator algebra, 245
Chapter 9 twisted generating function (new definition), 257 positive (resp., negative) part of generating function (new definition), 257 normal ordering, 257 twisted vertex operator, 258 quadratic differential operator, 264 general twisted vertex operator, 266 involution of &,' 274 *l-eigenspace of 8, 274 subspace of ' V of weight 2, 295
Chapter 10 finite set, 298 binary linear code, 298 dual code, 299 Mathieu group, 302 lattices associated with e, 303 Leech lattice, 304 &-root lattice, 305 Niemeier lattices, 308 Conway groups, 3 10 canonical &module, 3 13 moonshine module, 3 15 sum of untwisted and twisted modules, 315 Griess algebra, 316 affinization of Griess algebra, 317 centralizer of involution in M, 324 central involution in C, 325 graded character, 330 modular group, 336 modular invariant with zero constant term, 337
497
498
List of Frequently Used Symbols
Chapter 11
Q
Lo, L , , L
v, KO, vi#l,v,, VL,, VLl,
8
00 0 1
root lattice of type A?, 343 lattices associated with C?,343 untwisted spaces, 344 twisted spaces, 345 Lie algebra of type At4, 348 involution of 8 , 349 involution of W,, 349, 350, 364, 371
Chapter 12 24-element set, 375 binary Golay code, 375 lattices associated with A, 376 cosets associated with A, 376, 389 groups associated with lattices, 380, 383 spaces associated with a union of cosets M of Q, 385 automorphism of W and @, 396 automorphism of V , 397
6
Monster, 400 Q-form of Vh, 408 M-invariant positive definite Hermitian form on Vh, 41 I
Chapter 13 N H Ij
A
certain group, 420 certain group, 423 group associated with N(A?), 431 group associated with N(D,6),441
Appendix P -End, Hom R
infinite product of graded spaces, 462 spaces of linear operators from sums to products, 463, 464 analytic continuation, 466
L 2 n i ,[ dz
contour integral, 473