Trollrike/Forsmobo AB Sweden, 2009. - 536 p. - ISBN 978-969-9236-03-7, eBook, English.
(Abdus Salam School of Mathematical Sciences Lahore, Pakistan).
Summary. This volume contains.
(1) introduction into representation theory of simple Lie superalgebras (by J. Bernstein, P. Grozman, D. Leites, and I. Shchepochkina);
(2) introduction into representation theory of simple modular Lie superalgebras (by S. Bouarroudj, B. Clarke, P. Grozman, A. Lebedev, and D. Leites);
(3) examples of calculations feasible to be performed without computer (E. Poletaeva);
(4) formulation of open problems at course work, M.S. and Ph.D. theses level.
Most of the open problems are supposed to be solved with the help of a computeraided study.
Editor’s preface.
In this Volume 2, a sequel to the standard text-book material on the representation theory succinctly given in the first volume, I’ve collected the information needed to begin his or her own research. Mathematics is a language of sciences, and any language requires a dictionary.
This volume is an analog of an annotated dictionary: It contains few theorems but lots of notions. One should not try to learn all the examples of (simple) Lie superalgebras; for the first reading it is better to concentrate on the simplest examples and most profound notions (gl or sl, vect, q) and try to apply the rare theorems to these particular examples.Contents.
Editor’s preface.
Notation often used.
Part I Representations of Lie superalgebras over C.
Background over C (D. Leites).
Examples of simple Lie superalgebras and their relatives over C (D. Leites, I. Shchepochkina).
Invariant differential operators: solving O. Veblen’s problem (P. Grozman, D. Leites, I. Shchepochkina).
The Lie superalgebra osp(1|2), connections over symplectic manifolds and representations of Poisson algebras (J. Bernstein).
Poisson superalgebras as analogs of the general linear Lie algebra. The spinor and oscillator representations (D. Leites, I. Shchepochkina).
Irreducible representations of solvable Lie superalgebras (A. Sergeev).
How to realize Lie algebras by vector fields (I. Shchepochkina).
The analogs of Riemann and Penrose tensors on supermanifolds (E. Poletaeva).
Appendix. The dimension formula for irreducible sl(n)-modules.
Tables.
The nonholonomic Riemann and Weyl tensors for flag manifolds (P. Grozman, D. Leites).
Lie superalgebras of supermatrices of complex size (P. Grozman, D. Leites).
Symmetries wider than supersymmetry and simple Volichenko algebras (D. Leites).
Part II Modular Lie algebras and Lie superalgebras: Background and examples (A. Lebedev).
Background: The modular case.
Non-degenerate bilinear forms in characteristic 2, related contact forms, simple Lie algebras and Lie superalgebras.
g(A): Examples in characteristic 2.
Presentations of finite dimensional symmetric classical modular Lie algebras and superalgebras.
Analogs of the Hamiltonian, Poisson, and contact Lie superalgebras in characteristic 2.
Queerification.
Part III Modular Lie algebras and Lie superalgebras: Background (B. Clarke).
Decompositions of the tensor products of irreducible sl(2)-modules in characteristic 3 (B. Clarke).
Part IV Quest for simple modular Lie algebras and Lie superalgebras (S. Bouarroudj, P. Grozman, D. Leites).
Towards classification of simple finite dimensional modular Lie superalgebras (D. Leites).
Classification of simple finite dimensional modular Lie superalgebras with Cartan matrix (S. Bouarroudj, P. Grozman, D. Leites).
Selected problems (D. Leites).
References.
Index.
