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Carter R.W. — Simple groups of Lie type
Carter R.W. — Simple groups of Lie type



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Название: Simple groups of Lie type

Автор: Carter R.W.

Аннотация:

Since Chevalley showed in 1955 how to construct analogues of the complex simple Lie groups over arbitrary fields, these 'simple groups of Lie type' and their twisted analogues have been the subject of detailed investigation from a number of different points of view. This book is intended to serve as an introduction to the theory of Chevalley groups. It is not an exhaustive account of these groups, but concentrates on the basic results in the structure theory of the Chevalley groups and the twisted groups.
The Chevalley groups are studied in this book as groups of automorphisms of Lie algebras. This approach implies a concentration on the adjoint Chevalley groups—indeed the universal Chevalley groups and other isogenous groups have only been touched on in the development. In developing the theory we have found it necessary to assume a certain familiarity with the theory of simple Lie algebras over the complex field. Fortunately several good accounts of this theory are now available, and the information which we need about the simple Lie algebras has been collected together in a survey chapter...


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1972

Количество страниц: 346

Добавлена в каталог: 04.12.2010

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Предметный указатель
$(U^{-}_{w})^{1}$, the subgroup $U_{w}^{-}\cap G^{1}$ of a twisted group $G^{1}$      229
$(\mathfrak{I}_{J})_{n}$, the set of homogeneous polynomials in $\mathfrak{I}_{J}$ of degree n      140
$A_{l}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$A_{l}(K)$, the Chevalley group of type $A_{l}$ over K      64
$A_{l}(q)$, the Chevalley group of type $A_{l}$ over GF(q)      121
$A_{rs}$, the Cartan integer associated with a pair of roots      38
$B^{1}$, the subgroup $B\cap G^{1}$ of a twisted group $G^{1}$      230
$B_{l}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$B_{l}(K)$, the Chevalley group of type $B_{l}$ over K      64
$B_{l}(q)$, the Chevalley group of type $B_{l}$ over GF(q)      121
$C_{ijrs}$, the constants occurring in Chevalley's commutator formula      77
$C_{J}$, the element of the Coxeter complex associated with a set J of fundamental roots      31
$C_{l}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$C_{l}(K)$, the Chevalley group of type $C_{l}$ over K      64
$C_{l}(q)$, the Chevalley group of type $C_{l}$ over GF(q)      121
$d_{1},...,d_{l}$, the degrees of the basic polynomial invariants of W      130
$D_{J}$, the set of distinguished coset representatives of $W_{J}$ in W      29
$D_{l}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$D_{l}(K)$, the Chevalley group of type $D_{l}$ over K      64
$D_{l}(q)$, the Chevalley group of type $D_{l}$ over GF(q)      121
$E_{6}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$E_{6}(K)$, the Chevalley group of type $E_{6}$ over K      64
$E_{6}(q)$, the Chevalley group of type $E_{6}$ over GF(q)      121
$E_{7}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$E_{7}(K)$, the Chevalley group of type $E_{7}$ over K      64
$E_{7}(q)$, the Chevalley group of type $E_{7}$ over GF(q)      121
$E_{8}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$E_{8}(K)$, the Chevalley group of type $E_{8}$ over K      64
$E_{8}(q)$, the Chevalley group of type $E_{8}$ over GF(q)      121
$e_{r}$, a root vector in a simple Lie algebra      51
$F_{4}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$F_{4}(K)$, the Chevalley group of type $F_{4}$ over K      64
$F_{4}(q)$, the Chevalley group of type $F_{4}$ over GF(q)      121
$GL_{n}(K)$, the general linear group of degree n over K      2
$G^{1}$, a twisted group      226
$G_{2}$, the type of a simple Lie algebra over $\mathbb{C}$      43
$G_{2}(K)$, the Chevalley group of type $G_{2}$ over $K$      64
$G_{2}(q)$, the Chevalley group of type $G_{2}$ over GF(q)      121
$h(\chi)$, the automorphism of the Lie algebra $\mathfrak{L}_{K}$ determined by a K-character $\chi$ of P      98
$H^{1}$, the diagonal subgroup $H\cap G^{1}$ of a twisted group $G^{1}$      226
$h_{r}$, the co-root associated to a root r      42
$H_{r}$, the hyperplane orthogonal to a root r      20
$h_{r}(\lambda)$, the element of H associated with a root r and an element $\lambda$ of K      92
$I_{1},...,I_{l}$, a set of basic polynomial invariants of the Weyl group W      126
$L_{J}$, a Levi subgroup of a parabolic subgroup $P_{J}$      119
$m_{i},...,m_{l}$ the exponents of the Weyl group      155
$M_{r,s,i}$, the integer $\frac{1}{i!}N_{r,s}N_{r,r+s}...N_{r,(i-1)r+s}$      61
$N^{1}$, the monomial subgroup $N\cap G^{1}$ of a twisted group $G^{1}$      226
$N_{J}$, the inverse image in N of the subgroup $W_{J}$ of W      108
$n_{J}(w)$, the number of elements fixed by w in the W-orbit containing $C_{J}$ of the Coxeter complex      136
$N_{r,s}$, the structure constant of a simple Lie algebra determined by a pair of roots      52
$n_{r}$, the generator $x_{r}(1)x_{-r}(-1)x_{r}(1)$ of N      93
$n_{r}(t)$, the element $x_{r}(t)x_{-r}(-t^{-1})x_{r}(t)$ of N      96
$n_{w}$, an element of N mapping to w in W      115
$O_{2l}^{+}(q)$, the orthogonal group of degree 2l over GF(q) leaving invariant a quadratic form of index l      6
$O_{2l}^{-}(q)$, the orthogonal group of degree 2l over GF(q) leaving invariant a quadratic form of index l-1      6
$O_{J}$, the number of $\rho$-orbits in a $\rho$-invariant set J of fundamental roots      254
$O_{n}(K,f)$, the orthogonal group of degree n over K leaving invariant the quadratic form f      4
$PGL_{n}(K)$, the projective general linear group of degree n over K      2
$PSO_{n}(K,f)$, the projective special orthogonal group of degree n over K leaving invariant the quadratic form f      5
$PSp_{n}(K)$, the projective symplectic group of degree n over K      4
$PSU_{n}(K,f)$, the projective special unitary group of degree n over K leaving invariant the Hermitian form f      7
$P\Omega_{n}(K,f)$, the projective group of the commutator subgroup $\Omega_{n}(K,f)$ of $O_{n}(K,f)$      5
$p_{1},...,p_{l}$, a system of fundamental roots      38
$P_{J}$, the parabolic subgroup $BN_{J}B$ associated with a set J of fundamental roots      108
$P_{W^{1}}(t)$, the polynomial $\sum\limits_{w\in W^{1}} t^{l(w)}$      254
$P_{W}(t)$, the polynomial $\sum\limits_{w\in W} t^{l(w)}$      135
$q_{1},...,q_{l}$, the fundamental weights of a simple Lie algebra      98
$SL_{2}$      67 81 87 88
$SL_{n}(K)$, the special linear group of degree n over K      2
$SO_{n}(K,f)$, the special orthogonal group of degree n over K leaving invariant the quadratic form f      5
$Sp_{n}(K)$, the symplectic group of degree n over K      2
$SU_{n}(K,f)$, the special unitary group of degree n over K leaving invariant the Hermitian form f      7
$U^{+}_{w}$, the product of the root subgroups corresponding to positive roots transformed by w into positive roots      115
$U^{-}_{w}$, the product of the root subgroups corresponding to positive roots transformed by w into negative roots      115
$U^{1}$, the unipotent subgroup $U\cap G^{1}$ of a twisted group $G^{1}$      226
$U_{J}$, the unipotent radical of the parabolic subgroup $P_{J}$      119
$U_{m}$, the subgroup of U generated by the root subgroups corresponding to roots of height at least m      78
$U_{n}(K,f)$, the unitary group of degree n over K leaving invariant the Hermitian form f      7
$U_{r}$, the product of the root subgroups corresponding to positive roots other than r      104
$V^{1}$, the unipotent subgroup $V \cap G^{1}$ of a twisted group $G^{1}$      226
$V_{-r}$, the product of the root subgroups corresponding to negative roots other than -r      105
$W(\Sigma)$, the group of type-preserving automorphisms of the abstract Coxeter group $\Sigma$      284
$W^{1}$, the subgroup of elements of W commuting with the isometry $\tau$      217
$w^{J}_{0}$, the element of $W_{J}$ which transforms every positive root in $\Phi_{J}$ to a negative root      218
$w_{0}$, the element of W which transforms each positive root to a negative root      20
$W_{J}$, the subgroup of W generated by the fundamental reflections $w_{r}$ for r in J      27
$w_{r}$, the reflection in the hyperplane orthogonal to the root r      12
$X_{r}$, the root subgroup of a Chevalley group corresponding to the root r      68
$x_{r}(t)$, the generator exp (t ad $e_{r}$) of a Chevalley group      64
$X_{S}$, the subgroup of a Chevalley group generated by the root subgroups corresponding to roots in S      227
$X_{S}^{1}$, the root subgroup $X_{S}\cap G^{1}$ of a twisted group $G^{1}$      227
$\bar G$, the universal Chevalley group associated with G      190
$\bar Z$, the centre of the universal Chevalley group $\bar G$      190
$\chi$, a K-character of P      97
$\chi_{r,\lambda}$, the K-character of P taking value $\lambda^{A_{rs}}$ at the root s      48
$\epsilon_{1},...,\epsilon_{l}$, the eigenvalues of $\tau$ with eigenvectors $I_{1},...,I_{l}$      254
$\eta_{1},...,\eta_{l}$, the eigenvalues of the isometry $\tau$ of $\mathfrak{V}$      251
$\eta_{r,s}$, the integer $\pm 1$ defined by $n_{r}.e_{s}=\eta_{r}$, $_{s}e_{w_{r}(s)}$      93
$\hat G$, the extension of G by its group of diagonal automorphisms      118
$\hat H$, the group of automorphisms $h(\chi)$ of $\mathfrak{L}_{K}$      98
$\hat \mathfrak{I}$, the set of alternating polynomials on $\mathfrak{V}$      139
$\hat \mathfrak{I}_{n}$, the set of homogeneous alternating polynomials of degree n      140
$\hat \mathfrak{V}$, the dual space of $\mathfrak{V}$      123
$\lambda(r)$, an integer which takes value 1 if r is a short root and 2 or 3 if r is a long root      204 206
$\mathbb{C}$, the complex field      34
$\mathbb{Q}$, the field of rationals      153
$\mathbb{R}$, the real number field      12
$\mathbb{Z}$, the ring of rational integers      50
$\mathfrak{H}$, a Cartan subalgebra of a simple Lie algebra      35
$\mathfrak{H}_r$, the 1-dimensional subspace of $\mathfrak{H}$ containing $h_r$      83
$\mathfrak{I}$, the ring of polynomial invariants of a Weyl group W      124
$\mathfrak{I}^{+}$, the set of polynomials in $\mathfrak{I}$ with no constant term      125
$\mathfrak{I}_{J}$, the set of polynomials on $\mathfrak{V}$ invariant under $W_{J}$      140
$\mathfrak{I}_{n}$, the set of homogeneous polynomials in $\mathfrak{I}$ of degree n      132
$\mathfrak{L}$, a Lie algebra, usually simple over $\mathbb{C}$      33
$\mathfrak{L}(K)$, the Chevalley group of type $\mathfrak{L}$ over K      64
$\mathfrak{L}(q)$, the Chevalley group of type $\mathfrak{L}$ over GF(q)      121
$\mathfrak{L}_{K}$, the Lie algebra over K constructed from the simple Lie algebra $\mathfrak{L}$ over $\mathbb{C}$      62
$\mathfrak{L}_{r}$, the 1-dimensional subspace of $\mathfrak{L}$ corresponding to the root r      36
$\mathfrak{S}$, the algebra of polynomial functions on $\mathfrak{V}$      123
$\mathfrak{S}_{n}$, the set of homogeneous polynomials in $\mathfrak{S}$ of degree n      132
$\mathfrak{V}$, a vector space      2
$\mathfrak{V}^{1}$, the set of elements of $\mathfrak{V}$ invariant under the isometry $\tau$      217
$\mathfrak{V}_{J}$, the subspace of $\mathfrak{V}$ spanned by a set J of fundamental roots      27
$\nu(f)$, the Witt index of a form f      5
$\Omega$, a building      292
$\Omega(G;B,N)$, the building associated with a group G with (B, N)-pair      293
$\Omega_{n}(K,f)$, the commutator subgroup of the orthogonal group $O_{n}(K,f)$      5
$\Phi$, a root system      12
$\Phi^{*}$, the set of co-roots of roots in $\Phi$      49
$\Phi^{+}$, the set of positive roots in $\Phi$      14
$\Phi^{-}$, the set of negative roots in $\Phi$      16
$\Phi^{1}$, the set of projections on to $\mathfrak{V}^{1}$ of roots in $\Phi$      219
$\Phi_{J}$, the set of roots in $\Phi$ which lie in the subspace $\mathfrak{V}_{J}$      27
$\Pi$, a fundamental system of roots      13
$\Pi^{1}$, the set of projections on to $\mathfrak{V}^{1}$ of roots in $\Pi$      219
$\rho$, a symmetry of the Dynkin diagram      200
$\Sigma$, an abstract Coxeter complex      281
$\sigma$, an automorphism of a Chevalley group obtained by combining a graph and a field automorphism      225
$\Sigma(W,\Pi)$, the abstract Coxeter complex associated with a Weyl group W and fundamental system $\Pi$      288
$\succ$, a total order relation on $\mathfrak{V}$      13
$\tau$, the isometry of $\mathfrak{V}$ determined by a symmetry $\rho$ of the Dynkin diagram      201 217
$^{2}A_{l}(K)$, the twisted group of type $A_{l}$ over K      251
$^{2}A_{l}(q^{2})$, the twisted group of type $A_{l}$ over $GF(q^{2})$      251
$^{2}B_{2}(2^{2m+1})$, the twisted Suzuki group of type $B_{2}$ over $GF(2^{2m+1})$      251
$^{2}B_{2}(K)$, the twisted Suzuki group of type $B_{2}$ over K      251
$^{2}D_{l}(K)$, the twisted orthogonal group of type $D_{l}$ over K      251
$^{2}D_{l}(q^{2})$, the twisted orthogonal group of type $D_{l}$ over $GF(q^{2})$      251
$^{2}E_{6}(K)$, the twisted group of type $E_{6}$ over K      251
$^{2}E_{6}(q^{2})$, the twisted group of type $E_{6}$ over $GF(q^{2})$      251
$^{2}F_{4}(2^{2m+1})$, the twisted Ree group of type $F_{4}$ over $GF(2^{2m+1})$      251
$^{2}F_{4}(K)$, the twisted Ree group of type $F_{4}$ over K      251
$^{2}G_{2}(3^{2m+1})$, the twisted Ree group of type $G_{2}$ over $GF(3^{2m+1})$      251
$^{2}G_{2}(K)$, the twisted Ree group of type $G_{2}$ over K      251
$^{3}D_{4}(K)$, the triality twisted group of type $D_{4}$ over K      251
$^{3}D_{4}(q^{3})$, the triality twisted group of type $D_{4}$ over $GF(q^{3})$      251
$^{i}\mathfrak{L}(K)$, the twisted group constructed from a Chevalley group $\mathfrak{L}(K)$ by means of a symmetry of order i      251
(B,N)-pair      107—114 294
(B,N)-pair in twisted groups      227 230
(B,N)-pair, operating on a building      299 302
A, the rational group algebra of e(Q)      148
Abstract Coxeter complex      281 284 288 291 296
ad x, the adjoint map of left multiplication by the element x in a Lie algebra      34
Adjacent chambers      276
Adjoint Chevalley group      198
Alternating elements      148
Alternating polynomials      139
apartments      292 296
Automorphisms, of Chevalley groups      199 211
Automorphisms, of simple Lie algebras      60 63
B, the subgroup UH of a Chevalley group      104
Basic polynomial invariants      128 129
Betti numbers      169
Borel subgroup      104
Bruhat decomposition      104 106 109
Building      292 296 302
Building, associated with a group with (B,N)-pair      293
C, a chamber      21
Canonical form, for elements of a Chevalley group      115 117
Canonical form, for elements of a twisted group      229
Canonical form, for elements of a unipotent subgroup      78
Cartan decomposition      35
Cartan matrix      43—45 99 122
Central series of unipotent group      78
Chamber      21—23 275
Chamber complex      275
Chevalley basis      56
Chevalley group, definition      64
Chevalley's theorems, on commutator formula      76
Chevalley's theorems, on existence of integral basis      56
Chevalley's theorems, on polynomial invariants      128
Classification of simple Lie algebras      43
Closed set of roots      114
Co-roots      49
Coleman's theorem      166
Commutator formula      76
COMPLEX      274
Conway groups      308
Coxeter complex      30—32 136 137 255 288
Coxeter element      156
Coxeter group      25 284 286 292
Coxeter's theorem      156
Degrees of basic invariants      130—133 145 155
Derivations      34 61
Diagonal automorphism      200
Diagonal subgroup H, of a Chevalley group      97 99 117
Diagonal subgroup H, of a twisted group      238 244
Dihedral subgroup of Weyl group      158 161
Distinguished coset representatives      30 138
Double coset decomposition      109 110
Dual root system      49 50
Duality of exponents      156 168
Dynkin diagram, of a simple Lie algebra      40
Dynkin diagram, of the twisted groups      224
e(Q), a multiplicative group isomorphic to the additive group Q      148
Eigenvalues of Coxeter element      165 166 168
Elementary symmetric polynomials      124
Existence theorem for simple Lie algebras      42
Exponential map      60 66
Exponents of Weyl group      169
Extraspecial pair of roots      58
Factorization of the polynomial $\sum t^{l(w)}$, in Chevalley groups      135
Factorization of the polynomial $\sum t^{l(w)}$, in twisted groups      254
Field automorphism      200
Finite Chevalley groups      120—122
Finite twisted groups      251
Fischer groups      309
Folding      277
Frobenius — Perron theorem      162
Fundamental group      99
Fundamental reflection      17
Fundamental system of roots      13 19 21
Fundamental weights      98 146 148
G', the commutator subgroup (derived group) of G      170
G, a group, usually a Chevalley group      68
gallery      276
Generators and relations, for Chevalley group      190
Generators and relations, for Weyl group      23 25
GF(q), the Galois field with q elements      2
Graph automorphism, of the group $G_{2}(K)$      206
Graph automorphism, of the groups $B_{2}(K)$, $F_{4}(K)$      204
h(r), the height of a root r      16
H, the diagonal subgroup of a Chevalley group G      97
Height of a root      16 77 153—155
Held group      306
Higman — Sims group      307
Homomorphism, from $SL_{2}(K)$      88
Homomorphism, from $SL_{2}(\mathbb{C})$      87
Ideal of Lie algebra      33
Induced character      136
Isomorphism theorem for simple Lie algebras      42
J, a subset of the set $\Pi$ of fundamental roots      27
Jacobi identity      33
Jacobian      134 166
Janko groups      305
K*, the multiplicative group of non-zero elements of K      97
K, a field      2
K-character      97 99 121 199
K-character, self-conjugate      238
Killing form      34
l(w), the length of an element w of the Weyl group      18
l, the rank of a simple Lie algebra      35
Leech lattice      308
Lefshetz fixed point formula      169
length function      18 136
Levi decomposition      118
Levi subgroup      119
Lie algebra, definition      33
Linear groups      2 184 185
Lyons group      306
Macdonald's theorem      151
Mathieu groups      303
Matsumoto's Theorem      286
McLaughlin group      307
Monomial subgroup N, of Chevalley group      101
Monomial subgroup N, of twisted group      228
Monomial subgroup N, relationship to Weyl group      102
Morphism of chamber complexes      276
N, the monomial subgroup of a Chevalley group G      101
N, the number of positive roots      43
Nilpotent derivations      61 66 69
Nilpotent Lie algebra      35
Opposite folding      280
Order, of Coxeter elements      163 168
Order, of finite Chevalley groups      122
Order, of finite twisted groups      253 259 262
Order, of sporadic simple groups      310
Ordering of vector space      13
Orthogonal groups      4—8 184—188 271
P, the additive group generated by $p_{1},...,p_{l}$      97
Parabolic subgroups, of a Chevalley group      118 119
Parabolic subgroups, of a group with (B,N)-pair      111—113
Parabolic subgroups, of a twisted group      231
Parabolic subgroups, of a Weyl group      27—30 32
Poincare polynomial      169
Polynomial invariants of Weyl group      123
Positive systems of roots      13
Q, the additive group generated by $q_{1},..., q_{l}$      99
R, the root of maximum height      156
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