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Berezin F.A., Kirillov A.A. (ed.) — Introduction to Superanalysis
Berezin F.A., Kirillov A.A. (ed.) — Introduction to Superanalysis

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Название: Introduction to Superanalysis

Авторы: Berezin F.A., Kirillov A.A. (ed.)


The first three chapters were prepared for publication by Berezin himself and thus have been improved but slightly. They contain the basic facts on Grassmann algebras, differentiation and integration of functions of anticommuting variables, as well as on the advancement of the linear algebra technique in Z2-graded spaces. All this material is described in detail for a reader who is not familiar with the subject.
Then, according to the author's intention, the global theory of super-manifolds, the theory of Lie superalgebras and supergroups and their representations were to be discussed. Finally, the second part of the book was to be devoted to applications.
From all this material only Chapter 6, on Lie superalgebras, and a preliminary version of Chapter 4 devoted to supermanifolds, were found among Berezin's completed manuscripts. In addition to that, part of the book was explained by Berezin several years ago in a series of five preprints of the Institute of Theoretical and Experimental Physics (ITEP), Moscow.
For the present edition, Chapter 4 was completely rewritten by V. P. Palamodov. Its content goes beyond that sketched by Berezin and consists of new results (for instance, the construction of the example of a nonretractable complex supermanifold of dimensions (4,2)). Although the reader's mathematical knowledge is here assumed to be more advanced in comparison with the previous chapters, we hope that all the material presented is clearly explained and of interest to theoretical physicists.
The fifth chapter of the present book corresponds to the sixth chapter of Berezin's manuscript. The text was only slightly improved upon by the editor.
The second part of the book consists of the texts of the above-mentioned ITEP preprints supplemented by two appendices, one (Appendix II) having been written by V. I. Ogievetsky. The texts of the preprints are, in the main, unchanged. Only some paragraphs which are treated in the first part of the book were omitted and some misprints and inaccuracies corrected.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Издание: 1 edition

Год издания: 1987

Количество страниц: 424

Добавлена в каталог: 10.04.2010

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Предметный указатель
$Z_{2}$-graduation      166
$Z_{2}(Z_{+})$-graded K-algebra      166
$Z_{2}(Z_{+})$-graded Lie algebra      170 see
$Z_{2}(Z_{+})$-graded linear space      90
$Z_{2}(Z_{+})$-graded linear space, linear algebra in Grassmann envelope of      90
$\mathbb{K}$-algebra      137
$\mathbb{K}$-algebra, $\mathbb{Z}_{+}(\mathbb{Z}_{2})$-graded      166
$\mathbb{K}$-algebra, homomorphism of      137
$\mathbb{K}$-ringed space      139
$\mathbb{K}$-ringed space of the type $\mathscr{M}$      140
$\mathbb{K}$-ringed space, morphism of      140
Algebra      137
Algebra Lie, $\mathbb{Z}_{2}$-graded      170
Algebra Lie, with Grassmann structure      177
Algebra of operators in spaces $\mathbb{K}^{p, q}$, $\mathbb{K}^{p, q}(\Lambda)$      119
Algebra of Poisson brackets      184 219—228
Algebra over field $\mathbb{K}$      29 131
Algebra with convergence      31
Algebra with unit      30
Algebra, $L(p|\mathbb{C})$      30
Algebra, $L(p|\mathbb{R})$      30
Algebra, $Mat(p,q|\Lambda)$      95
Algebra, $\Lambda(U)$      55
Algebra, $\Lambda_{p, q}(U)$      74
Algebra, $\Lambda_{p, q}(U)$, differentiation and integration of elements of      74
Algebra, $\mathfrak{U}$      29 119
Algebra, $\mathfrak{U}$, absolutely irreducible      119
Algebra, $\mathfrak{U}$, graded irreducible      119
Algebra, $\mathfrak{U}$, irreducible      119
Algebra, $\mathfrak{U}(\Sigma)$      31
Algebra, $\mathfrak{U}^{M}$      31
Algebra, $\mathfrak{U}^{p, q}$      137
Algebra, $\mathscr{A}(U)$      30
Algebra, $\mathscr{E}^{p}$      137
Algebra, $\mathscr{H}^{p, q}$      138
Algebra, $\mathscr{H}^{p}$      137
Algebra, $\mathscr{P}(U)$      30
Algebra, A(U)      3
Algebra, algebraic set of generators of      31
Algebra, associative      29
Algebra, automorphism of      33
Algebra, commutative      29
Algebra, complex      29
Algebra, coset of      32
Algebra, enveloping      279
Algebra, finite-dimensional      29
Algebra, Grassmann      29 35
Algebra, homomorphisms of      33
Algebra, infinite-dimensional      29
Algebra, isomorphisms of      33
Algebra, linear, in $\mathbb{Z}_{2}$-graded spaces      90
Algebra, Mat(p,q)      95
Algebra, real      29
Algebra, special basis of      120
Algebra, system of generators of      31
Algebra, topological set of generators of      31
Algebra, Z-($Z_{+}$-)graded      166
Antiautomorphism      46
Automorphism of algebras      33
Automorphism of algebras, semisimple      46
Automorphism of algebras, the main linear part of      45
Base of a Lie supergroup      247
Basis of open sets      4
Berezinian      ix 82
Bundle, conormal      156
Burnside's theorem      122
Cartan dual to the Lie superalgebra      172
Cartan generators      259
Cartan subalgebra      237
Cartan subalgebra of type I (II)      237
Category      131
Category of $\mathbb{K}$-ringed spaces of the type $\mathscr{M}$      140
Centre of enveloping algebra      279
Character of supergroup representation      307
Chart      140
Complex envelope of real Lie superalgebra      171
Construction of a fibre space from a presheaf      133
Construction of a sheaf from a fibre space      134
Construction of a sheaf from a presheaf      134
Contact transformations      213
Coordinates, Grassmann canonical      252
Coordinates, Grassmann polar (angular, radial)      264
Coordinates, local      3 259
Coordinates, neighbourhoods      3
Coordinates, polar (angular, radial)      260
Coordinates, transformations      5
Coset of algebra      32
Degree of element in Grassmann algebra      37
Derivative with respect to anticommuting variable      74
Diffeomorphism      62
Diffeomorphism, leaving invariant an external form      204
Diffeomorphism, leaving invariant an integral form      202
Dual (Cartan)      172
Element, degree of      37
Element, even (odd)      37 54
Element, homogeneous      34 37
Element, of $\Lambda_{q}$, independent      52
Enveloping algebra      279
Enveloping algebra, centre of      279
Factor algebra      33
Factor algebra of Grassmann algebra      52
Fibre of the sheaf      134
Fibre of the sheaf, space      134
Fibre product of $Z_{2}$-graded $\mathbb{R}$-ringed space      152
Form differential      16
Form, exterior      16
Form, integral      202
Function of anticommuting variables      36
Function with support      79
Function, Grassmann algebra-valued      55
Function, Grassmann analytic      60
Function, theorem of implicit      66
Generating set of ideals      38
Generators of $\Lambda_{p, q}(U)$      60
Generators of $\Lambda_{p, q}(U)$, even (odd)      61
Generators of $\mathfrak{U}_{p, q}(U)$      5 258
Generators of algebra      31
Generators, canonical      35
Generators, Cartan      259
Generators, extendible system of      64
Generators, extension of the system of      35
Geometry, Hamiltonian      17
Geometry, symplectic      17
Germ      134
Grassmann algebra      29 35
Grassmann algebra, antiautomorphism of      46
Grassmann algebra, automorphisms of      44
Grassmann algebra, canonical generators of      35
Grassmann algebra, dimension of      36
Grassmann analytic continuation      246 253
Grassmann analytic extension of function      60
Grassmann analytic functions algebra      60 255
Grassmann analytic representation algebra      255
Grassmann canonical coordinates      252
Grassmann conjugate of the Lie algebra $\mathscr{G}(N)$      281
Grassmann envelope of $Z_{2}$-graded linear space      92
Grassmann envelope of algebra L(p, q)      94
Grassmann envelope of irreducible algebra      3 123
Grassmann envelope of Lie superalgebra      177 213 237
Grassmann envelope of space $\mathbb{K}^{p, 0}$, $\mathbb{K}^{0, q}$, $\mathscr{K}^{p, q}$      92
Grassmann polar (radial, angular) coordinates      262
Group of operators in spaces $\mathbb{K}^{p, q}$, $\mathbb{K}^{p, q}(\Lambda)$      119
Group, $Aut^{0}_{A}(\Lambda)$      44
Group, $Aut^{1}_{A}(\Lambda)$      44
Group, $Aut_{n}(\Lambda)$      45
Group, $C(m, n|\Lambda)$      129
Group, $C_{1}(m, n|\Lambda)$      129
Group, $GL(p, q|\Lambda)$      123
Group, $GL(p|\Lambda)$      123
Group, $GL(\mathbb{C}||p, q|\Lambda)$      124
Group, $Osp(p|\Lambda)$      130
Group, $Osp(\mathbb{C}||p, q|\Lambda)$      124
Group, $Osp(\mathbb{R}||p, q|\Lambda)$      127 129
Group, $Osp_{\mathbb{R}^{*}}(p, q|\Lambda)$      127
Group, $SL(p, q|\Lambda)$      123
Group, $SL(p|\Lambda)$      123
Group, $SL(\mathbb{C}||p, q|\Lambda)$      124
Group, $SU(p, q|\Lambda)$      124
Group, $SU(p|\Lambda)$      130
Group, $U(p, q|\Lambda)$      124
Group, $U(p|\Lambda)$      130
Group, complex orthosymplectic      125
Group, semidirect product of      173
Hamiltonian field      204
Hamiltonian function of a vector field      204
Homomorphism of $\mathbb{K}$-algebras      137
Homomorphism of Lie supergroups      248
Homomorphism of restriction      58
Homomorphism of rings      136
Homomorphism, canonical      137
Homomorphism, kernel of      34
Homomorphism, structural      137
Ideal      32
Ideal, left (right)      32 136
Ideal, maximal      137
Ideal, two-sided      32 136
Ideals, generating set of      34
Inductive limit      134
Inner product in $\mathscr{G}$, $\mathscr{G}(N)$      28
Integral on the Lie supergroup      267
Integral with respect to anticommuting variables      77
Integration by parts      77
Integration in the case of functions without support      87
Integration over supermanifolds      11
Integration, change of variables      78
Involution of the first (second) kind      47 50
Isomorphism of algebras      31
Kernel of homomorphism      34
Laplace operators      231 283
Laplace operators, generalized      305
Laplace operators, radial part of      231 284 312
Lie algebra with Grassman structure      176
Lie algebra, $\mathbb{Z}_{2}$-graded      170
Lie algebra, semidirect product of      175
Lie derivatives      194
Lie group with Grassmann structure      177
Lie superalgebra      19 170 231
Lie superalgebra of contact transformations      213
Lie superalgebra of type I (II)      237
Lie superalgebra with Grassmann structure      176
Lie superalgebra, $osp(\mathbb{C}|2p, q)$      182
Lie superalgebra, $osp(\mathbb{K}|n)$      185
Lie superalgebra, $osp(\mathbb{R}|2p, 2q^{'}+q^{''})$      184
Lie superalgebra, $osp(\mathbb{R}|2p, p)$      182
Lie superalgebra, $S_{L}(4, N)$      186 189
Lie superalgebra, $u_{E}(4, N)$      188
Lie superalgebra, $\mathscr{D}_{p, q}(U)$      191
Lie superalgebra, adjoint (coadjoint) representation of      180 216 280
Lie superalgebra, C(m, n)      240
Lie superalgebra, Cartan dual to      172
Lie superalgebra, commutative      171
Lie superalgebra, complex of envelope of real      171
Lie superalgebra, examples of      180
Lie superalgebra, gl(N)      190
Lie superalgebra, gl(p, q)      239
Lie superalgebra, Grassmann envelope of      177 213 238
Lie superalgebra, integral on      270
Lie superalgebra, L(p, q)      180
Lie superalgebra, L(Q|p, q)      189
Lie superalgebra, nilpotent      171
Lie superalgebra, reductive      171
Lie superalgebra, representations of      21 181
Lie superalgebra, semidirect product of      175
Lie superalgebra, semisimple      171
Lie superalgebra, simple      171
Lie superalgebra, sl(p, q)      181
Lie superalgebra, solvable      171
Lie superalgebra, su(p, q)      181
Lie superalgebra, u(2, 2, N)      186
Lie superalgebra, u(p, q)      181
Lie supergroup      19 245
Lie supergroup, GL(m, n)      248
Lie supergroup, global      248
Lie supergroup, homomorphisms of      248
Lie supergroup, invariants (Laplace operators) of      22 307
Lie supergroup, Lie operators on      260
Lie supergroup, representations of      21 250 307 353
Linear algebra in $Z_{2}$-graded spaces      90
Linear operators      90
Linear operators in $Z_{2}$-graded spaces      94
Linear space, $Z_{2}$-graded      90 see
Liouville theorem      109
Manifold      3
Manifold, complex-analytic      142
Manifold, dimensionality      3
Manifold, Grassmann analytical      10
Manifold, k-dilferentiable      3 142
Mapping, transition      141
Mapping, underlying      138
Matrices with anticommuting elements      99
Model spaces, category of      140
Modules      35
Modules, $O_{z^{-}}$      154
Modules, left (right)      35
Modules, locally free $O_{z^{-}}$      155
Modules, two sided      35
Morphism of $\mathbb{K}$-ringed space      140
Morphism of presheaves      132
Morphism of ringed space      138
Morphism of sheaves      135
Morphism of type $\mathscr{M}$      141
Morphism, fibre product of      153
Neighbourhood, kth infinitesimal      154
Obstruction to the extension of a $Z_{+}$-graduation      168
Obstruction, first      156 159
Obstruction, higher      163
Obstruction, kth      164
Operator, $\circ\atop\rho^{V}_{U}$      3
Operator, Laplace (— Casimir)      231 283
Operator, Laplace (— Casimir), generalized      305
Operator, Laplace (— Casimir), radial part of      231 284
Operator, Lie, on Lie supergroups      260
Operator, linear, in $Z_{2}$-graded space      94
Operator, odd (even)      94
Parity of $\Lambda$      55
Parity of element of Grassmann algebra      38
Parity, automorphism      43
Particle spin dynamics      374
Poincare — Bilkhoff — Witt theorem      279
Poisson brackets      205
Poisson brackets, algebra of      221
Poisson brackets, invariant      222
Polynomial, continuable      293
Polynomial, left (right)      34
Preimage of a sheaf      136
Presheaf      132
Presheaf, morphisms of      132
Product, direct of objects of a category      150
Product, direct of supermanifolds      150
Product, fibre      152
Product, inner      281
Product, semidirect of Lie algebras      175
Product, semidirect of Lie superalgebras      175
Queer determinant      113
Queer trace      113
Rank of $\mathscr{O}_{z}$-module      155
Representation of Lie algebra $\mathscr{G}(\Lambda)$      215
Representation of superalgebras      21 180
Representation of superalgebras, adjoint (coadjoint)      180 216 280
Representation of supergroups      21 248
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