Kock J. — Frobenius Algebras and 2-D Topological Quantum Field Theories :: Электронная библиотека попечительского совета мехмата МГУ

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Kock J. — Frobenius Algebras and 2-D Topological Quantum Field Theories

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Название: Frobenius Algebras and 2-D Topological Quantum Field Theories

Автор: Kock J.

Аннотация:

Throughout, the emphasis is on the interplay between algebra and topology with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering the new quantum areas of mathematics, who will learn a host of modern techniques that will prove useful for future work.

Язык:

Рубрика: Физика/Квантовая теория поля/Теория струн/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 238

Добавлена в каталог: 18.09.2005

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Предметный указатель

-module      88
-set      169
-algebra      86
-algebra, in graphical terms      110
$\beta$ , Frobenius pairing      95 111
$\Delta$ , category of finite ordinals      178
$\Delta$ , category of finite ordinals, generators and relations      183
$\Delta$ , category of finite ordinals, graphical description      180
$\Delta$ , comultiplication      116
$\eta$ , unit      109
$\mathbb{N}$ , natural numbers      143-145
$\mathbf{2Cob}$ , category of 2-cobordisms      59-77 171
$\mathbf{2Cob}$ , category of 2-cobordisms, generators      62
$\mathbf{2Cob}$ , category of 2-cobordisms, objects      59
$\mathbf{2Cob}$ , category of 2-cobordisms, relations      69
$\mathbf{2Cob}$ , category of 2-cobordisms, skeleton      61
$\mathbf{2Cob}$ , category of 2-cobordisms, universal property      217
$\mathbf{2TQFTf}_{k}$ , category of 2-dimensional TQFTs      168
$\mathbf{Ab}$ , category of abelian groups      201
$\mathbf{Algf}_{k}$ , category of -algebras      88 203
$\mathbf{cAlgf}_{k}$ , category of commutative -algebras      203
$\mathbf{Cat}$ , category of all categories and functors      227
$\mathbf{cFA}_{k}$ , category of commutative Frobenius algebras      132 173
$\mathbf{cFrob(V)}$ , category of commutative Frobenius objects in $\mathbf{V}$       216
$\mathbf{FA}_{k}$ , category of Frobenius algebras      132
$\mathbf{FinOrd}$ , category of finite ordered sets      177
$\mathbf{FinSet}$ , category of finite sets      188
$\mathbf{FinSet}_{0}$ , category of finite sets and bijections      192
$\mathbf{Frob(V)}$ , category of Frobenius objects in $\mathbf{V}$       215
$\mathbf{gr-cAlg}_{k}$ , category of graded-commutative -algebras      203
$\mathbf{grVect}_{k}$ , category of graded vector spaces      166
$\mathbf{Mod}_{R}$ , category of -modules      160
$\mathbf{Mon(V)}$ , category of monoids in $\mathbf{V}$       198
$\mathbf{MonCat}$ , category of monoidal categories      153
$\mathbf{Mon}$ , category of monoids (in Set)      142
$\mathbf{nCob}$ , category of -cobordisms      45
$\mathbf{nTQFT}_{k}$ , category of -dimensional TQFTs      168
$\mathbf{n}$ , th ordinal      179
$\mathbf{Repr}_{k}(V)$ , category of linear representations of a symmetric monoidal category $\mathbf{V}$       167
$\mathbf{Ring}$ , category of rings      201
$\mathbf{rMod}_{A}$ , category of right -modules      89
$\mathbf{SymMonCat}$ , category of symmetric monoidal categories      166
$\mathbf{Vect}_{k}$ , category of -vector spaces      79
$\mathbf{X}$ , category of ‘planar 2-cobordisms’      216
$\mathcal{B}_{k}$ , braid group      195
$\mathcal{S}_{k}$ , the symmetric group      57 193
$\mu$ , multiplication map      109
$\Phi$ , category of finite cardinals      189
$\Phi$ , category of finite cardinals, generators and relations      194
$\Phi$ , category of finite cardinals, graphical description      191
$\Sigma$ , category of finite cardinals and bijections      192
$\varepsilon$ , Frobenius form      94 111
(co)commutative (co)algebra      122
(co)commutativity, relation in $\mathbf{2Cob}$       69
2-dimensional TQFTs and Frobenius algebras      171-175
Algebra      86
Associativity, implied by the Frobenius relation      119
Associativity, in a -algebra      86
Associativity, in a category      223
Associativity, in a monoid      141 197
Associativity, of a pairing of -modules      91
Associativity, relation in $\mathbf{2Cob}$       69
B, braid category      196
Bialgebra (digression)      135 214
Bilinear pairing      81
Bimonoid (digression)      214
Boundary      11
Braiding      169
Cardinal sum      189
Cartesian product of categories      149
Cartesian product of sets      139
Category      223
Classification of surfaces      64
Coalgebra      106 213
Coassociativity, of a coalgebra      107
Coassociativity, relation in $\mathbf{2Cob}$       69
Cobordism class      26
Cobordism groups (digression)      27
Cobordism, 1-dimensional      21 23-25
Cobordism, oriented      22
Cobordism, unoriented      18
Cohomology rings      104
Commutative monoid      143
Commutative monoid, in a symmetrical monoidal category      200
Comonoid      212
Composition, notation      xv
Composition, of cobordism classes      43
Comultiplication      107
coordinates      123
Copairing      81
Coproduct (categorical)      231
Counit      107 212
Critical point      16
Crossed -set      170
Crossing-over      192
Cylinder      12 25 29
Cylinder, as identity cobordism      44
Decomposition of a cobordism      28
Degeneracy map      185
Direct sum      56 93 130 176
Disjoint union      49-52
Division algebras      99
Dual of a vector space      80
Dual of an -module      89
Duality in a Frobenius algebra      131
Eckmann-Hilton argument      211
Endomorphism monoid      145
Equivalence of categories      229
Equivalence of cobordisms      26
Essentially surjective      229
Euler characteristic      63
Face map      185
Faithful      229
Fibre product      231
Field extensions      99
Finite cardinal      189
Finite ordered set      177
Finite ordinal      178
Free monoid      143
Frobenius      101
Frobenius algebra      94 215
Frobenius algebra, commutative      122 173
Frobenius algebra, examples      98-104
Frobenius algebra, symmetric      97
Frobenius algebra, trivial      99
Frobenius algebras and 2-dimensional TQFTs      171-175

Frobenius form      94
Frobenius form, in graphical terms      111
Frobenius object (in a monoidal category)      215
Frobenius pairing      95
Frobenius pairing, in graphical terms      111
Frobenius relation      117 215
Frobenius relation, in $\mathbf{2Cob}$       70
Frobenius relation, in coordinates      127
full      229
functor      226
Functor category      228
Generators and relations for $\Delta$       183-187
Generators and relations for $\mathbf{2Cob}$       62
Generators and relations for $\Phi$       194
Generators and relations for a category      58
Generators and relations for a monoidal category      61
Generators and relations for the braid group $\mathcal{B}_{k}$       196
Generators and relations for the symmetric group $\mathcal{S}_{k}$       57
Genus      63
Gluing      35-43
Gluing, cobordisms      42
Gluing, cylinders      40
Gluing, intervals      38
Gluing, topological spaces      35
Gorenstein rings      102
Graded algebra      203
Graded algebra, graded vector space      166
Graded algebra, graded-commutative algebra      203
Graphical calculus      108
Group      143 225
Group algebras      100 137
group characters      102
Half-space      11
Handle operator      77
Historical remarks      55 101 120 175
Hopf algebras (digression)      136
Identity cobordism      44
In-boundary      14
Initial object      230
Interchange law      211
interval      12 15
Interval, as cobordism      23
Jacobian algebras      103
Jacobson radical      100
Koszul symmetry      166
Linear functional (linear form)      80
Linear representation of a symmetrical monoidal category      167
Manifold with boundary      11
Matrix algebras      99
Metric      125
Monoid      141
Monoid action      146
Monoid, commutative      143 200
Monoid, in $\Delta$       204
Monoid, in $\Phi$       208
Monoid, in a monoidal category      197
Monoid, trivial      144 203
Monoidal category      150
Monoidal category, braided      169
Monoidal category, nonstrict      154-157
Monoidal category, symmetric      160
Monoidal functor      153
Monoidal functor, nonstrict      156
Monoidal functor, on $\Delta$       205
Monoidal functor, symmetric      166
Monoidal natural transformation      167
Moore’s Theorem      57
Morse function      17
movies      19
Multiplication in a -algebra      86
Natural transformation      227
Natural transformation, monoidal      167
Nondegenerate pairing      81
Normal form (for a connected surface)      64
Ordered set, order-preserving map      177
Ordinal sum      179
Orientation      13
Out-boundary      14
Pairing      81
Pairing, associative      91
Pairing, nondegenerate      81
Paralleling      57 151
Positive normal      14
Product (categorical)      230
Pushout      232
Regular interval theorem      41
Regular representation      101
Semi-monoid      148
Semi-simple algebra      100 106
Simplicial identities      186
Simplicial set      187
Singleton category      149
singleton set      140
Skeleton      59
Skew-fields      99
Snake decomposition      29
Snake relation (nondegeneracy of a pairing)      112
Strictification theorem      157
Sufficiency of the relations (in $\mathbf{2Cob}$ )      73
Sweedler’s trigonometric coalgebra      107 128
Symmetric Frobenius algebra      97
Symmetric Frobenius algebra, graphical      123
Symmetric monoidal category      160
Symmetric monoidal functor      166
Symmetric monoidal functor, on $\mathbf{nCob}$       168
Symmetric monoidal functor, on $\Phi$       209
Symmetry      160-166 174
Tensor calculus      123
Tensor product      80
Tensor product, of (co)algebras      132
Tensor product, of Frobenius algebras      134
Terminal object      230
Three-point function      113 125
Topological quantum field theory (TQFT)      30 54
Topologist’s delta      187
Trigonometric coalgebra of Sweedler      107 128
Trivial Frobenius algebra      99
Twist cobordism      52 61 72-77
Twist map      121 140
U-tube      26
Unit in a -algebra      86
Unit in a monoid      141 197
Vector space      79
Vector space, graded      166
‘Main theorem’      173
‘Main theorem’, generalised      217

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