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Bouc S. — Green Functors And G-Sets
Bouc S. — Green Functors And G-Sets



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Название: Green Functors And G-Sets

Автор: Bouc S.

Аннотация:

This book provides a definition of Green functors for a finite group G, and of modules over it, in terms of the category of finite G-sets. Some classical constructions, such as the associated categroy or algebra, have a natural interpretation in that framework. Many notions of ring theory can be extended to Green functors (opposite Green functor, bimodules, Morita theory, simple modules, centres, ...). There are moreover connections between Green functors for different groups, given by functors associated to bisets. Intended for researchers and students in representation theory of finite groups it requires only basic algebra and category theory, though knowledge of the classical examples of Mackey functors is probably preferable.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 342

Добавлена в каталог: 02.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$.^V$      181
$A \times B$      46
$a \times m$      47 76
$A(X^2)$      81
$a\circ_X m$      72
$a\circ_Y a'$      65
$A\otimes K$      278
$A^{op}$      127
$A_Y.j$      115
$b \hat{\circ} a$      125
$C_M(L)$      143
$C_M(\alpha)$      143
$FP_{H, V}^G$      296
$FQ_{H, V}^G$      296
$f_*, f^*$      68
$f_H^G$      308
$F_M$      71
$G-set\downarrow_X$      53
$Green_R(G)$      47
$G\U.\alpha$      183
$H^\oplus(-, R)$      97
$k_1(L)$      183
$k_2(L)$      183
$L_{X, V}$      275
$Mack_R(G)$      7
$M\circ U$      168
$M\hat{\otimes}_BN$      146
$M\mapsto M_Y$      8
$M^o$      130
$M_F$      72
$m_{(E, \epsilon)}$      194
$N\circ U_{|A}$      237 256
$p_1(L)$      183
$p_2(L)$      183
$S^\phi$      206
$S_\phi$      206
$S_{H, V}^G$      282
$S_{X, V}$      275
$T^\phi$      193
$T_\phi$      193
$Z\natural T$      242
$[m\otimes n]_K$      14
$[m\otimes n]_{(Y, \phi)}$      16
$\circ_H$      167
$\delta_{X_1, ..., X_n}^{U}$      171
$\epsilon_H^G$      308
$\eta_Z$      185
$\hat{A}(H)$      276
$\hat{C}$      310
$\hat{\iota}_H^G$      218
$\iota_H^G$      218
$\kappa_{S, T}^U$      230
$\lambda_Z$      199
$\left( \begin{array}{c} x \\ f(x) \end{array} \right)$      50
$\mathcal{D}_U(X)$      186
$\mathcal{H}(M, N)$      9
$\mathcal{H}_A(M, N)$      145
$\mathcal{H}_A(M, N)$ of internal homomorphisms      11
$\mathcal{H}_A(M, N)$ of tensor product (first)      15
$\mathcal{H}_A(M, N)$ of tensor product (second)      23
$\mathcal{I}_X$      316
$\mathcal{L}(M_1, ..., M_n;P)$      29
$\mathcal{L}_U(M)$      193
$\mathcal{L}_U(\theta)$      196
$\mathcal{Q}_U(M)$      188
$\mathcal{R}_U(M)$      207
$\mathcal{R}_U(\theta)$      209
$\mathcal{S}_U(M)$      205
$\mu_1(G)$      335
$\nu$-disjoint      188
$\nu_{(Y, f)}$      185
$\omega$      84
$\omega_N(K)$      218
$\Omega_Y$      112
$\overline{M}(H)$      273
$\overline{N}_G(H)$      278
$\pi$-perfect      308
$\pi_R(G)$      307
$\pi_Z$      199
$\Sigma_\pi (G)$      313
$\tau_{X, Y}^U$      168
$\times^U$      171
$\times^{op}$      127
$\times_H$      181
$\underline{M}(H)$      273
$\varepsilon_A$      134
$\varepsilon_{A\circ U}$      171
$\varepsilon_{\mathcal{L}_U(A)}$      232
$\widetilde{X}$      194
$\zeta_A(H)$      315
$\zeta_A(X)$      316
${}^U \times$      231
A-mod      48
a.m      49
Adjunction between $\hat{\otimes}$ and $\mathcal{H}$      38
Adjunction between $\hat{\otimes}_B$ and $\mathcal{H}_A$      148
Adjunction, co-unit      185
Adjunction, unit      185
Algebra associated to a Green functor      84 99 164
Alperin’s conjecture      304
B      52
Balanced      99 154
Bifunctoriality      46 47
Bimodule      141
Bimodule construction      153
Bimodule, structure on $\mathcal{H}{M, N)$      142
Biproducts      68
Biset      167
Biset, composition with      168
Blocks of Mackey algebra      335
Burnside functor as Green functor      55
Burnside functor as initial object      57
Burnside functor as Mackey functor      52
Burnside functor as unit      59
Cartesian product in $\mathcal{C}_A \times \mathcal{C}_A$      109
Cartesian product in $\mathcal{C}_A$      103
Category $\mathcal{D}_U(X)$      186
Category, adding direct summands to a      310
Category, associated to a Green functor      67
Category, equivalence of      79 310—312 314
Category, representation of      71 81
Centre      305
Centre of Yoshida algebra      335
Coinduction      180
Coinflation      168 217
Commutant      143
Commutative Green functor      110 129 305
Commute      136
Composition and associated categories      173
Composition and Green functors      170
Composition and modules      175
Composition and tensor product      168
Composition with a biset      168
Direct summand in $\mathcal{C}_A$      82
Divides in $\mathcal{C}_A$      82
Dual of a module      130
Embedding of algebras      112
Endosimple A-module      304
Equivalence of categories      79 84 112 129 141 155 310—312 314
Equivalence of the definitions of Green functors      48
Examples of algebras $A(\Omega^2)$      94 95
Examples of composition with a biset      168
Examples of Frobenius morphisms      227
Examples of functors $\mathcal{L}_U(M)$ and $\mathcal{R}_U(M)$      215
Examples of functors $\zeta_A$      332
Examples of Green functors $\mathcal{L}_U(A)$      264
Finitely generated module over a Green functor      114
Frobenius morphisms      223
Functor from G-sets to $\mathcal{C}_A$      68
Functor of evaluation      81
Functorial ideal      291
Functoriality of $M\hat{\otimes}_B N$      147
Functoriality of $U\mapsto U \circ_G X$      177
Functoriality of $\mathcal{H}(M, N)$ and $M\hat{\otimes}$      24
Functoriality of $\mathcal{H}_A(M, N)$      146
G.s.t      229
Generators and relations for $A(\Omega^2)$      85 88
Generators and relations for the Mackey algebra      7
Green functors and solvable $\pi$-subgroups      314
Green functors, centre      305
Green functors, composition with a biset      170
Green functors, definition in terms of G-sets      46
Green functors, definition in terms of subgroups      41
Green functors, direct sum      305
Green functors, tensor product      134
G\U.Y      183
Identification of $\mathcal{H}_A(M, N)$      145
Identification of $\zeta_A$      320 323 327
Injective on the orbits      177 186 192
Left adjoint and Morita contexts      273
Left adjoint and tensor product      272
Left adjoint to $M\mapsto M\circ U$      197
Left adjoint to $Z\mapsto U\circ_H Z$      183
Lindner construction      94
Mackey algebra      7
Mackey algebra, anti-automorphism      10
Mackey axiom      5
Mackey functors, composition with a biset      168
Mackey functors, definition as modules      7
Mackey functors, definition in terms of G-sets      6
Mackey functors, definition in terms of subgroups      5
Mackey functors, internal homomorphisms      9
Mackey functors, tensor product of      9
Module of finite type      114
Module over a Green functor      41 47
Module over a Green functor, examples      61
Module over the Burnside functor      59
Module, dual      130
Module, right      129
Morita context      100 154
Morita equivalence      99
Morita equivalence of algebras $A(X^2)$      99 100
Morita Theory      155 160
Morphism of bimodules      141
Morphism of Green functor      41
Morphism of mackey functors      5 6
Morphism of modules over a Green functor      42
Morphism, bilinear      37 127
Morphism, Frobenius      223
Morphism, n-linear      29
Morphism, universal property      29
Multiple of, in CA      82
Natural transformation      316
Non-commutative tensor product      164
Opposite Green functor      127
Product $\circ_H$      167
Product $\hat{\circ}$      125
Product $\times$ for $A\hat{\otimes}B$      134
Product $\times$ for $\mathcal{H}{M, M)$      123
Product $\times$ for $\mathcal{L}_U(A)$      231
Product $\times$ for $\zeta_A$      317
Progenerator      115 117 118 155 161 321
Projective G-algebra      292
Projective relative to      102
Projective relative to, solvable $\pi$-subgroups      314
q(L)      183
Relative projectivity      100
Representation of a category      71 81
Residual rings      274
Restriction      5
Restriction for $M\hat{\otimes} N$      15 23
Restriction for $\mathcal{H}{M, N)$      10
Right adjoint to $M\mapsto M\circ U$      210
Right modules      129
S.T      229
Same stabilizers      100
Simple Green functors      291
Simple Green functors, classification      295
simple modules      275
Simple modules, classification      276
Simple modules, structure      282
Source algebra      114
Surjective Morita context      154
Tensor product and composition with a biset      168
Tensor product and left adjoints      227
Tensor product and residues      298
Tensor product and right adjoints      242
Tensor product of "algebras" over $b(\Omega^2)$      164
Tensor product of Green functors      134
Tensor product of Green functors, universal property      137
Tensor product of modules over Green functors      140
Tensor product of simple modules      298
Tensor product, associativity      38
Tensor product, commutativity      38
Tensor product, n-fold      25
Tensor product, n-fold, universal property      29
Thevenaz’s theorem      295
Trace      see "Transfer"
TRANSFER      5
Transfer for $M\hat{\otimes}N$      15 23
Transfer for $\mathcal{H}{M, N)$      10
U.Y      183
Unitary Green functor      46
Unitary morphism of Green functors      41 43
U[X]      173
Yoshida algebra      95
Yoshida algebra, centre      335
Z(A)      305
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