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Krause G.R., Lenagan T.H. Growth of Algebras and Gelfand-Kirillov Dimension
Krause G.R., Lenagan T.H.  Growth of Algebras and Gelfand-Kirillov Dimension









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: Growth of Algebras and Gelfand-Kirillov Dimension

: Krause G.R., Lenagan T.H.

:

During the two decades that preceded the publication of the first edition of this book, the Gelfand-Kirillov dimension had emerged as a very useful and powerful tool for investigating non-commutative algebras. At that time, the basic ideas and results were scattered throughout various journal articles. The first edition of this book provided a much-needed reliable and coherent single source of information. Since that time, the book has become the standard reference source for researchers. For this edition, the authors incorporated the original text with only minor modifications. Errors have been corrected, items have been rephrased, and more mathematical expressions have been displayed for the purpose of clarity. The newly added Chapter 12 provides broad overviews of the new developments that have surfaced in the last few years, with references to the literature for details. The bibliography has been updated and accordingly, almost double the size of the original one. The faithful revision and contemporary design of this work offers time-honored expertise with modern functionality: A keenly appealing combination. So, whether for the classroom, the well-tended mathematical books collection, or the research desk, this book holds unprecedented relevance.


: en

: /

:

:

ed2k: ed2k stats

: revised edition

: 1999

: 212

: 09.03.2011

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Algebra, almost commutative      61 73
Algebra, associated $\mathbb{N}^{n}$-graded      182
Algebra, associated graded      64
Algebra, associated graded, noetherian      70
Algebra, Auslander Gorenstein      187
Algebra, Auslander-regular      187
Algebra, catenary      112 188
Algebra, central simple      126
Algebra, Cohen Macaulay      188
Algebra, commutative      39
Algebra, Filtered      64
Algebra, filtered, multi-      181
Algebra, finite dimensional      7
Algebra, finite dimensional, locally      14
Algebra, finitely generated      5
Algebra, free      7
Algebra, GK-partitive      190
Algebra, graded      61
Algebra, graded, connected      61
Algebra, graded, connected $\mathbb{N}^{n}$-      180
Algebra, graded, finitely      61
Algebra, graded, homogeneous element of      61
Algebra, group      139
Algebra, Heisenberg      75
Algebra, Lie      see Lie algebra
Algebra, locally finite      180
Algebra, monomial      157
Algebra, normally separated      188
Algebra, PI      see Pi-algebra
Algebra, polynomial      9
Algebra, polynomial identity      125 191
Algebra, prime      48
Algebra, primitive      162
Algebra, quotient      39
Algebra, representable      191
Algebra, semi-commutative      179
Algebra, semiprime      49
Algebra, somewhat commutative      179
Algebra, strongly finitely presented      172
Algebra, strongly finitely presented, locally      172
Algebra, symmetric      67
Algebra, universal enveloping      7 109
Algebra, Weyl      26 87 183
Algebra, Weyl, quotient division algebra of      100
Annihilator      33
Basss formula      151
Bergmans Gap Theorem      18 155
Bernstein class      89
Bernstein number      78
Bernsteins inequality      88 189
Bimodule      54
Catenarity      see Algebra catenary
Center      39
cl. Kdim      see Krull dimension classical
Common denominator property      38
Commutator subgroup      140
Derivation      24
Derivation, $\sigma$-      164
Derivation, locally nilpotent      42 169
Derivation, locally triangulizable      41 169
Differential operator      87
Dimension of level q      154
Dimension, bandwidth      155
Dimension, faithful      191
Dimension, filter      189
Dimension, Gelfand Kirillov      14 51
Dimension, ideal invariant      56
Dimension, Krull, classical      40
Dimension, Krull, Gabriel Rentschler      33
Dimension, q-      154
Dimension, super      13 154
Dimension, symmetric      56
Dimension, uniform      33
distribution      95
Dixmier map      112 187
Eigenvalue      41
Endomorphism, locally algebraic      164
Endomorphism, locally nilpotent      41
Endomorphism, locally triangulizable      41
Equivalent filtrations      see Filtration of module
Essential      see Module essential
Exact      see Gelfand Kirillov dimension
f-generating set      see Group nilpotent
f-growth      see Group nilpotent
f-length      see Group nilpotent
Faithful dimension      see Dimension faithful
Filter dimension      see Dimension filter
Filtration of algebra      64
Filtration of algebra, discrete      64
Filtration of algebra, finite      64
Filtration of algebra, multi-      181
Filtration of module      64
Filtration of module, discrete      64
Filtration of module, equivalent      68
Filtration of module, finite      64
Filtration of module, multi-      182
Filtration of module, standard      67
Finite uniform dimension      33
Free semigroup      15
Free semigroup, order ideal in      18
Free semigroup, word in      15
Free semigroup, word in, minimal period of      16
Free semigroup, word in, periodic      16
Function, analytic      95
Function, growth of      see Growth of function
Function, meromorphic      95
Function, periodically polynomial      175
Function, rational      174
Function, return      189
Function, test      95
g-stable ideal      110
Gelfand Kirillov Conjecture      2
Gelfand Kirillov dimension      1
Gelfand Kirillov dimension of algebra      14
Gelfand Kirillov dimension of algebra, lower      166
Gelfand Kirillov dimension of algebra, upper      166
Gelfand Kirillov dimension of bimodule      55
Gelfand Kirillov dimension of module      51
Gelfand Kirillov dimension of module, exact      53 69 73 131 136 172 191
Generating subspace of algebra      5
Generating subspace of module      51
GKdim      see Gelfand Kirillov dimension
Goldie conditions      see Ring Goldie
Goldies Theorem      33 48
Grade      see Module grade
Grading of algebra      61
Grading of module      62
Graph      156
Graph, adjacency matrix of      156
Graph, chain      156
Graph, cycle      156
Graph, growth      156
Graph, overlap      158
Graph, overlap, faithful      159
Graph, path      156
Graph, path, cyclic      156
Graph, path, length of      156
Graph, Poincare series of      156
Graph, Ufnarovskii      157
Gromovs Theorem      139 196
Group, algebra      139
Group, finitely generated      139
Group, fundamental      140
Group, growth      see Growth of group
Group, linear      151
Group, nilpotent      139
Group, nilpotent, f-generating set      147
Group, nilpotent, f-growth      147
Group, nilpotent, f-length      147
Group, nilpotent, length of filtration      147
Group, nilpotent-by-finite      145
Group, polycyclic      142
Group, polycyclic, Hirsch number of      143
Group, solvable      139
Growth      1
Growth of algebra      6
Growth of function      6
Growth of group      140
Growth of group, exponential      140
Growth of group, intermediate      196
Growth of group, polynomial      140
Growth of group, subexponential      196
Growth of module      51
Growth, curve      155
Growth, exponential      6
Growth, intermediate      153
Growth, logarithmic      154
Growth, O(g(n))-      155
Growth, polynomial      6
Growth, subexponential      6 153
Height      see Prime ideal height
Heisenberg group      145 197
Hilbert series      159
Hilbert Samuel polynomial      68 76 78
Hodge algebra      19
Holonomic      see Module holonomic
Homogeneity      see Module homogeneous
Jacobson radical      126
Jacobson radical, nilpotent      126
Kaplanskys Theorem      126
Kdim      see Krull dimension
Krull dimension, classical      40
Krull dimension, Gabriel Rentschler      33 81
LD      see Transcendence degree lower
Ld-stable      186
Leading submodule      65
Lie algebra, $ad_{\mathfrak{g}}$-nilpotent subalgebra of      44
Lie algebra, algebraic      116
Lie algebra, enveloping algebra of      44 67 109
Lie algebra, finite dimensional      41
Lie algebra, infinite dimensional      7
Lie algebra, nilpotent      112 119
Lie algebra, semisimple      109
Lie algebra, solvable      44 84 109
localization      37
Localization at Ore sets arising from locally triangulizable inner derivations      169
Localization, central      127 167
Localization, normal      168
Markovs theorem      195
Module, associated $\mathbb{N}^{n}$-graded      182
Module, associated graded      64
Module, associated graded, noetherian      70
Module, essential submodule of      33
Module, filtered      64
Module, filtered, multi-      182
Module, grade of      187
Module, graded      62
Module, graded, $\mathbb{N}^{n}$-      180 182
Module, graded, finitely      62
Module, graded, homogeneous element of      62
Module, holonomic      89 183 191
Module, homogeneous      60 113
Module, multiplicity of      178
Module, Noetherian      65
Module, pure with respect to grade      187
Module, smooth      59
Module, uniform      33
Multi-filtration      181
Multi-filtration, finite      182
Multiplicity      see Bernstein number
Nilpotent, group      see Group nilpotent
Nilpotent, ideal      57 172
Nilpotent, Lie algebra      see Lie algebra nilpotent
Nilpotent, radical      57 173
Noether Normalization Theorem      40
Normal element      115 168
Normal element, local      168
Order, admissible      181
Order, good      181
Ore, condition      37
Ore, extension      24 164
Ore, set      37
Ore, set, of normal elements      168
Partitive      see Algebra GK-partitive
Periodically polynomial function      175
PI-algebra      18 125
PI-algebra, finitely generated      127
PI-algebra, noetherian      131
PI-algebra, prime      126
PI-algebra, simple      126
PI-degree      130
Poincare series of filtered algebra      174
Poincare series of graded algebra      174
Poincare series of graph      156
Poincare series of group      197
Poincare series of monomial algebra      159
Poincare series, multi-variable      180
Poincare series, nonrational      197
Poincare series, rational      156 174176 181 183
Poincare Birkhoff Witt, extension      163
Poincare Birkhoff Witt, Theorem      67
Polynomial identity      125
Polynomial, central      126
Polynomial, Hilbert      see Hilbert Samuel polynomial
Polynomial, Hilbert Serre      see Hilbert Samuel polynomial
Posners theorem      126
Prime ideal of enveloping algebra of solvable Lie algebra      110
Prime ideal, height of      34 109
Primitive ideal      109
Pure      see Module pure
Quillens Lemma      88
Quotient ring      see Ring of fractions
Rank      137
Rank of abelian group      169
Rank, reduced      137
Rational function      174
Regular element      33
Regular element, E(x)-      169
Return function      see Function return
Ring of differential operators      87
Ring of fractions      38
Ring, catenary      112
Ring, Goldie      33
Ring, irreducible      134
Ring, Krull symmetric      116
Ring, Noetherian      33
Ring, polynomial      26
Ring, power series      26
Ring, primary      134
Ring, prime      35
Ring, right FBN      173
Ring, semiprime      48
Ring, skew polynomial      164
Ring, skew-Laurent polynomial      144 190
Ring, skew-Laurent power series      101
Sensitive multiplicity condition      165
Superdimension      see Dimension super
Tauvels height formula      109
Tdeg-stable      184
Tensor product      28 165
Transcendence degree of a field      40
Transcendence degree of PI-algebra      126
Transcendence degree, Gelfand Kirillov      106 167 183
Transcendence degree, lower      185
Volume difference inequality      185
Width of multi-index      104
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