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Giambruno A., Zaicev M. — Polynomial Identities and Asymptotic Methods
Giambruno A., Zaicev M. — Polynomial Identities and Asymptotic Methods

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Название: Polynomial Identities and Asymptotic Methods

Авторы: Giambruno A., Zaicev M.

Аннотация:

This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory. A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of codimensions of any PI-algebra has integral exponential growth - the PI-exponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent. Results are extended to graded algebras and algebras with involution. The book concludes with a study of the numerical invariants and their asymptotics in the class of Lie algebras. Even in algebras that are close to being associative, the behavior of the sequences of codimensions can be wild. The material is suitable for graduate students and research mathematicians interested in polynomial identity algebras.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 352

Добавлена в каталог: 12.12.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$Aut^{\ast}(A)$      65
$C(k,n)^{mult}$      125
$c_n(a)$      88
$c_n^G(A)$      256
$c_n^{(h_1,...,h_n)}(A)$      98
$c_n^{gr}(A)$      268
$c_n^{sup}(A)$      283
$c_n^{\ast}(A)$      284
$c_n^{\mathbb{Z}_2}(A)$      273
$C_{a{p_m}}$      13
$C_{T_\lambda}$      49
$deg_{x_i}f$      1
$D_\lambda$      47
$End_A(M)$      29
$e_\lambda$      47
$e_{T_\lambda}$      49
$FS_n$      46
$F\{\xi\}$      12
$f{\equiv}0$      2
$F{\langle}X,Tr{\rangle}$      122
$F{\langle}X,{\ast}{\rangle}$      69
$F{\langle}X{\mid}G{\rangle}$      65
$F{\langle}X{\rangle}$      1
$F{\langle}X{\rangle}^{gr}$      66
$F{\langle}X{\rangle}^{\ast}$      56
$F{\langle}Y,Z{\rangle}$      69
$GKdim(A)$      39
$G^{gr}$      289
$G{\wr}S_n$      257
$Id^G(A)$      66
$Id^{gr}(A)$      66
$Id^{\ast}(A)$      284
$I_\lambda$      47
$L_k(x;y)$      129
$l_n(A)$      90
$MT_n$      122
$M_k(F)$      11
$M_k(G)$      83
$M_n(F{\oplus}cF)$      75
$M_{k,l}(F)$      75
$M_{k,l}(G)$      83
$M_{k{\times}l}(F)$      251
$M{\hat\otimes}N$      50
$PT_n$      122
$P_n$      53
$P_n(A)$      54
$P_n(\mathcal{V})$      54
$P_n^G$      256
$P_n^\ast$      284
$P_n^{gr}$      268
$P_n^{\mathbb{Z}_2}$      283
$P_n^{\mathbb{Z}_2}(A)$      273
$P_{r,n-r}$      273
$R_{T_\lambda}$      49
$St_m(x_1,...,x_m)$      13
$T_2$-ideal      80
$T_\lambda$      47
$UT(d_1,...,d_m)$      24
$UT_2(F)$      88
$UT_2^{gr}$      289
$UT_n(F)$      2
$\alpha{\models}n$      50
$\ast-exp(A)$      284
$\ast-var(A)$      284
$\chi_n^G(A)$      265
$\chi_n^{mtr}(M_k(F))$      135
$\chi_\lambda$      46
$\chi_{n_1,...,n_k}(A)$      266
$\chi{\downarrow}H$      46
$\chi{\uparrow}G$      46
$\Delta(\xi_1,...,\xi_{k^2})$      125
$\hat G$      63
$\lambda{\vdash}n$      15
$\mathbb{Z}_2-exp(A)$      276
$\mathcal{A}_{x_1,...,x_r}$      18
$\mathcal{V}$      4
$\mathcal{V}^\ast$      82
$\phi$-ideals      73
${\langle} S {\rangle}_T$      4
${\langle}\lambda{\rangle}$      264
ADX      309
Algebra of generic elements      11
Algebra of generic matrices      12
Algebra of invariants      124
Algebra, algebraic of bounded degree      14
Algebra, center      31
Algebra, supercommutative      83
Algebra, verbally prime      83
Amitsur      35 36 104 106 108 215 242—245
Amitsur identity      109
Amitsur trick      100
Amitsur — Levitzki theorem      18
Antiautomorphism      65
Artinian ring      29
AT-algebra      317
Aut(A)      63
Berele      38 117 212 274
Birkhoff theorem      4
Block-triangular matrix algebra      24
Branching theorem      50
C(k,n)      125
Capelli identity      13
Capelli polynomial      13
Central localization      34
CHARACTER      45
Character, induced      46
Character, inner product      45
Character, irreducible      45
Cocharacter      54
Cocharacter, $\ast$-cocharacter      284
Cocharacter, G-cocharacter      265
Cocharacter, graded cocharacter      238
Cocharacter, mixed trace cocharacter      135
Cocharacter, pure trace cocharacter      123
Codimension      88
Codimension, $\ast$-codimension      264 284
Codimension, $\mathbb{Z}_2$-codimension      273 284
Codimension, G-codimension      256
Codimension, graded codimension      268 283
Codimension, supercodimension      283
Codimension, trace codimension      123
Colength      90
Column-stabilizer      49
Commuting ring      29
Decomposable monomial      259
deg f      1
Dense set      29
det(a)      2
Dilworth      94
Discriminant      125
Drensky      20 28 137 185 187 207
Element, skew-symmetric      69
Element, symmetric      69
Elementary symmetric function      15
Essential G-identity      262
Essential hook      241
exp(A)      144
exp(f)      215
EXPONENT      144
Exponent of a polynomial      215
Exponent, $\ast$-exponent      284
Exponent, $\mathbb{Z}_2$-exponent      276
Exponent, superexponent      284
First Fundamental Theorem      125
Formanek      27 28 40 128 134 137
Free algebra with G-action      65
Free algebra with involution      69
Free algebra with trace      122
Free associative algebra      1
Free G-graded algebra      66
Free Lie algebra      310
Free superalgebra      69
Free supercommutative algebra      83
Frobenius reciprocity      46
g-identity      66
Gelfand — Kirillov dimension      37
Generalized square      244
Generic division ring      40
Generic element      11
Generic matrix      12
Graded algebra      5 61
Graded algebra, $\mathbb{Z}_2$-graded algebra      65
Graded algebra, G-graded algebra      61
Graded identity      66
Graded subalgebra      61
Graded subspace      61
Grassmann algebra      2 90
Grassmann envelope      81
GSPI-algebra      316
Gurevich      125
H(d,l)      58
h(d,l,t)      57
Halpin      28
Herstein      29
Homogeneous component      61
Homogeneous element      61
Hook      58
Hook formula      48
Hook number      48
Hook Theorem      105
Id(A)      3
Idempotent      45
Idempotent central      45
Idempotent essential      49
Idempotent minimal      45
Idempotent minimal graded      194
Indecomposable monomial      259
Induced module      46
Involution      69
Involution, exchange      77
Involution, symplectic      77
Involution, transpose      77
Jacobson      29
Kaplansky      27
Kaplansky's Theorem      31
Kasparian      28
Kemer      20 83 110 112 113 169
Koshlukov      20
Kostant      120
Krull dimension      39
Lattice permutation      51
Latyshev      94
Lewin Theorem      21
Lie algebra      307
Lie algebra, abelian      308
Lie algebra, adjoint representation      309
Lie algebra, center      309
Lie algebra, nilpotent      308
Lie algebra, representation      309
Lie algebra, simple      308
Lie algebra, solvable      308
Lie algebra, universal enveloping algebra      309
Lie commutator      2
Lie ideal      9
Lie identity      310
Littlewood — Richardson rule      51
Lower exponent      144
Maschke's Theorem      44
Mishchenko      184 268 289
Mixed trace polynomial      122
Monomial      1
Multilinearization process      7
Multipartition      264
Newton's formulas      15
Noether Normalization Theorem      39
Outer tensor product      50
Partition      46
Partition, conjugate      47
Permutation, d-bad      94
Permutation, d-good      94
PI-algebra      2
PI-exponent      144
Polynomial growth      171
Polynomial identity      2
Polynomial, alternating      12
Polynomial, central      26
Polynomial, consequence      7
Polynomial, equivalent      7
Polynomial, G-polynomial      66
Polynomial, homogeneous      5
Polynomial, linear      7
Polynomial, multialternating      138
Polynomial, multihomogeneous      5
Polynomial, multihomogeneous component      6
Polynomial, multilinear      7
Posner's theorem      34
Power sums symmetric function      15
Prime ring      34
Primitive ring      29
Procesi      27 40 123 125
Product of varieties      327
Pure trace polynomial      122
Razmyslov      19 20 27 28 123
Razmyslow — Kemer — Braun Theorem      35
Regev      94—96 108 117 128 141 184 212
Relatively free algebra      4
Representation      43
Representation, completely reducible      44
Representation, equivalent      44
Representation, irreducible      44
Representation, left regular      44
Robinson — Schensted correspondence      102
Rosset      18
Row insertion algorithm      101
Row-stabilizer      49
Rowen      33
s(A)      84
schur      120
Schur's lemma      29
Second Fundamental Theorem      125
Semiprime ring      32
Semistandard tableau      51
Sibirskii      125
Skew-tableau      51
Skolem — Noether theorem      78
SPI-algebra      314
Splitting field      30 45
Stable identity      10
Standard Lie polynomial      311
Standard polynomial      13
Standard tableau      47
Strip Theorem      107
Subdirect product      33
Superalgebra      65
Superalgebra, minimal      194
Superalgebra, reduced      240
Superenvelope      84
Supervariety      80
supexp(A)      284
Symmetric algebra      124
Symmetric function      15
Symmetric polynomial      15
T-ideal      3
T-ideal, verbally prime      82
tr deg(K/F)      39
tr(a)      2
Trace identity      122
Trace polynomial      122
Transcendence degree      39
Trivial grading      62
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