Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
Название: Introduction to Superanalysis
Авторы: Berezin F.A., Kirillov A.A. (ed.)
Аннотация:
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.
-graduation166 -graded K-algebra166 -graded Lie algebra170see -graded linear space90 -graded linear space, linear algebra in Grassmann envelope of90 -algebra137 -algebra, -graded166 -algebra, homomorphism of137 -ringed space139 -ringed space of the type 140 -ringed space, morphism of140 Algebra137 Algebra Lie, -graded170 Algebra Lie, with Grassmann structure177 Algebra of operators in spaces , 119 Algebra of Poisson brackets184219—228 Algebra over field 29131 Algebra with convergence31 Algebra with unit30 Algebra, 30 Algebra, 30 Algebra, 95 Algebra, 55 Algebra, 74 Algebra, , differentiation and integration of elements of74 Algebra, 29119 Algebra, , absolutely irreducible119 Algebra, , graded irreducible119 Algebra, , irreducible119 Algebra, 31 Algebra, 31 Algebra, 137 Algebra, 30 Algebra, 137 Algebra, 138 Algebra, 137 Algebra, 30 Algebra, A(U)3 Algebra, algebraic set of generators of31 Algebra, associative29 Algebra, automorphism of33 Algebra, commutative29 Algebra, complex29 Algebra, coset of32 Algebra, enveloping279 Algebra, finite-dimensional29 Algebra, Grassmann2935 Algebra, homomorphisms of33 Algebra, infinite-dimensional29 Algebra, isomorphisms of33 Algebra, linear, in -graded spaces90 Algebra, Mat(p,q)95 Algebra, real29 Algebra, special basis of120 Algebra, system of generators of31 Algebra, topological set of generators of31 Algebra, Z-(-)graded166 Antiautomorphism46 Automorphism of algebras33 Automorphism of algebras, semisimple46 Automorphism of algebras, the main linear part of45 Base of a Lie supergroup247 Basis of open sets4 Berezinianix82 Bundle, conormal156 Burnside's theorem122 Cartan dual to the Lie superalgebra172 Cartan generators259 Cartan subalgebra237 Cartan subalgebra of type I (II)237 Category131 Category of -ringed spaces of the type 140 Centre of enveloping algebra279 Character of supergroup representation307 Chart140 Complex envelope of real Lie superalgebra171 Construction of a fibre space from a presheaf133 Construction of a sheaf from a fibre space134 Construction of a sheaf from a presheaf134 Contact transformations213 Coordinates, Grassmann canonical252 Coordinates, Grassmann polar (angular, radial)264 Coordinates, local3259 Coordinates, neighbourhoods3 Coordinates, polar (angular, radial)260 Coordinates, transformations5 Coset of algebra32 Degree of element in Grassmann algebra37 Derivative with respect to anticommuting variable74 Diffeomorphism62 Diffeomorphism, leaving invariant an external form204 Diffeomorphism, leaving invariant an integral form202 Dual (Cartan)172 Element, degree of37 Element, even (odd)3754 Element, homogeneous3437 Element, of , independent52 Enveloping algebra279 Enveloping algebra, centre of279 Factor algebra33 Factor algebra of Grassmann algebra52 Fibre of the sheaf134 Fibre of the sheaf, space134 Fibre product of -graded -ringed space152 Form differential16 Form, exterior16 Form, integral202 Function of anticommuting variables36 Function with support79 Function, Grassmann algebra-valued55 Function, Grassmann analytic60 Function, theorem of implicit66 Generating set of ideals38 Generators of 60 Generators of , even (odd)61 Generators of 5258 Generators of algebra31 Generators, canonical35 Generators, Cartan259 Generators, extendible system of64 Generators, extension of the system of35 Geometry, Hamiltonian17 Geometry, symplectic17 Germ134 Grassmann algebra2935 Grassmann algebra, antiautomorphism of46 Grassmann algebra, automorphisms of44 Grassmann algebra, canonical generators of35 Grassmann algebra, dimension of36 Grassmann analytic continuation246253 Grassmann analytic extension of function60 Grassmann analytic functions algebra60255 Grassmann analytic representation algebra255 Grassmann canonical coordinates252 Grassmann conjugate of the Lie algebra 281 Grassmann envelope of -graded linear space92 Grassmann envelope of algebra L(p, q)94 Grassmann envelope of irreducible algebra3123 Grassmann envelope of Lie superalgebra177213237 Grassmann envelope of space , , 92 Grassmann polar (radial, angular) coordinates262 Group of operators in spaces , 119 Group, 44 Group, 44 Group, 45 Group, 129 Group, 129 Group, 123 Group, 123 Group, 124
Group, 130 Group, 124 Group, 127129 Group, 127 Group, 123 Group, 123 Group, 124 Group, 124 Group, 130 Group, 124 Group, 130 Group, complex orthosymplectic125 Group, semidirect product of173 Hamiltonian field204 Hamiltonian function of a vector field204 Homomorphism of -algebras137 Homomorphism of Lie supergroups248 Homomorphism of restriction58 Homomorphism of rings136 Homomorphism, canonical137 Homomorphism, kernel of34 Homomorphism, structural137 Ideal32 Ideal, left (right)32136 Ideal, maximal137 Ideal, two-sided32136 Ideals, generating set of34 Inductive limit134 Inner product in , 28 Integral on the Lie supergroup267 Integral with respect to anticommuting variables77 Integration by parts77 Integration in the case of functions without support87 Integration over supermanifolds11 Integration, change of variables78 Involution of the first (second) kind4750 Isomorphism of algebras31 Kernel of homomorphism34 Laplace operators231283 Laplace operators, generalized305 Laplace operators, radial part of231284312 Lie algebra with Grassman structure176 Lie algebra, -graded170 Lie algebra, semidirect product of175 Lie derivatives194 Lie group with Grassmann structure177 Lie superalgebra19170231 Lie superalgebra of contact transformations213 Lie superalgebra of type I (II)237 Lie superalgebra with Grassmann structure176 Lie superalgebra, 182 Lie superalgebra, 185 Lie superalgebra, 184 Lie superalgebra, 182 Lie superalgebra, 186189 Lie superalgebra, 188 Lie superalgebra, 191 Lie superalgebra, adjoint (coadjoint) representation of180216280 Lie superalgebra, C(m, n)240 Lie superalgebra, Cartan dual to172 Lie superalgebra, commutative171 Lie superalgebra, complex of envelope of real171 Lie superalgebra, examples of180 Lie superalgebra, gl(N)190 Lie superalgebra, gl(p, q)239 Lie superalgebra, Grassmann envelope of177213238 Lie superalgebra, integral on270 Lie superalgebra, L(p, q)180 Lie superalgebra, L(Q|p, q)189 Lie superalgebra, nilpotent171 Lie superalgebra, reductive171 Lie superalgebra, representations of21181 Lie superalgebra, semidirect product of175 Lie superalgebra, semisimple171 Lie superalgebra, simple171 Lie superalgebra, sl(p, q)181 Lie superalgebra, solvable171 Lie superalgebra, su(p, q)181 Lie superalgebra, u(2, 2, N)186 Lie superalgebra, u(p, q)181 Lie supergroup19245 Lie supergroup, GL(m, n)248 Lie supergroup, global248 Lie supergroup, homomorphisms of248 Lie supergroup, invariants (Laplace operators) of22307 Lie supergroup, Lie operators on260 Lie supergroup, representations of21250307353 Linear algebra in -graded spaces90 Linear operators90 Linear operators in -graded spaces94 Linear space, -graded90see Liouville theorem109 Manifold3 Manifold, complex-analytic142 Manifold, dimensionality3 Manifold, Grassmann analytical10 Manifold, k-dilferentiable3142 Mapping, transition141 Mapping, underlying138 Matrices with anticommuting elements99 Model spaces, category of140 Modules35 Modules, 154 Modules, left (right)35 Modules, locally free 155 Modules, two sided35 Morphism of -ringed space140 Morphism of presheaves132 Morphism of ringed space138 Morphism of sheaves135 Morphism of type 141 Morphism, fibre product of153 Neighbourhood, kth infinitesimal154 Obstruction to the extension of a -graduation168 Obstruction, first156159 Obstruction, higher163 Obstruction, kth164 Operator, 3 Operator, Laplace (— Casimir)231283 Operator, Laplace (— Casimir), generalized305 Operator, Laplace (— Casimir), radial part of231284 Operator, Lie, on Lie supergroups260 Operator, linear, in -graded space94 Operator, odd (even)94 Parity of 55 Parity of element of Grassmann algebra38 Parity, automorphism43 Particle spin dynamics374 Poincare — Bilkhoff — Witt theorem279 Poisson brackets205 Poisson brackets, algebra of221 Poisson brackets, invariant222 Polynomial, continuable293 Polynomial, left (right)34 Preimage of a sheaf136 Presheaf132 Presheaf, morphisms of132 Product, direct of objects of a category150 Product, direct of supermanifolds150 Product, fibre152 Product, inner281 Product, semidirect of Lie algebras175 Product, semidirect of Lie superalgebras175 Queer determinant113 Queer trace113 Rank of -module155 Representation of Lie algebra 215 Representation of superalgebras21180 Representation of superalgebras, adjoint (coadjoint)180216280 Representation of supergroups21248