Felix Alexandrovich Berezin (auth.), A. A. Kirillov (eds.)9789027716682, 9027716684
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. |
Table of contents :
Front Matter….Pages i-xii
Introduction….Pages 1-25
Front Matter….Pages 27-27
Grassmann Algebra….Pages 29-73
Superanalysis….Pages 74-89
Linear Algebra in Z 2 -Graded Spaces….Pages 90-130
Supermanifolds in General….Pages 131-169
Lie Superalgebras….Pages 170-228
Front Matter….Pages 229-229
Lie Superalgebras….Pages 231-244
Lie Supergroups….Pages 245-278
Laplace-Casimir Operators (General Theory)….Pages 279-311
The Radial Parts of the Laplace Operators on the Lie Supergroups U(p,q) and C(m, n) ….Pages 312-343
Construction of Representations of Lie Supergroups U ( p,q ) and C ( m,n )….Pages 344-373
Back Matter….Pages 374-424 |
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