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Dualities and representations of Lie superalgebras. (English) Zbl 1271.17001
Graduate Studies in Mathematics 144. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9118-6/hbk). xvii, 302 p. (2012).
This book gives a detailed account of the representation theory of finite-dimensional complex Lie superalgebras. To this extent, three main classes are considered in detail: the general linear superalgebras $$\mathfrak{gl}(m|n)$$, the orthosymplectic superalgebras $$\mathfrak{osp}(m|n)$$ and the queer superalgebras $$\mathfrak{q}(n)$$. The general results obtained for these classes can then be adapted for the remaining superalgebras of classical type. Among the various interesting points of the approach chosen by the authors, we emphasize the new proof presented for various well-known structural results in the literature. To this extent, minor inaccuracies already known to the specialist have been carefully corrected and presented in closed form. The main part of the book is devoted to the study of duality theories, focusing on the Schur, Howe and super-dualities. As the latter type has only been developed in recent years, this work constitutes one of the first systematic approaches to the topic.
The book is divided into six chapters, with a certain amount of technical appendices. The first chapter deals with the basic theory of Lie superalgebras, presenting the main notions as simplicity, invariant bilinear forms, root systems and Weyl groups. The important question of non-conjugacy of positive systems is considered in detail, leading to the notion of odd reflections. The chapter finishes with a description of the PBW theorem for superalgebras and some elementary facts of highest weight theory.
Chapter two deals with the three classes of superalgebras described above. Using odd reflections, the simple finite-dimensional modules are classified, in order to determine precisely the images of the corresponding Harish-Chandra homomorphism. Extremal weights are described by means of Young-tableaux, a procedure similar to the Lie algebra case, but not completely straightforward due to the existence of non-conjugate Borel subalgebras and the fact that the Weyl group in the superalgebra case plays a minor role in their representation theory.
The Schur duality is considered in chapter three. The authors begin with some general results and properties of associative superalgebras, and then progressively develop the theory. In this context, the Schur-Sergeev duality for general linear superalgebras is proved, hence providing a classification of irreducible polynomial modules. This enables to describe the characters using super-Schur polynomials. The duality for the queer superalgebra is considered in detail, leading also to an interesting description.
Chapter four, concerning invariant theory, can be seen as a preparatory compilation of results for the Howe duality, which is analyzed in chapter 5. First of all, the exact relation between the classical Lie superalgebras and the Weyl-Clifford algebras is established. The following paragraphs study the Howe duality for the $$A$$ and $$Q$$ types, the symplectic and orthogonal case and the infinite dimensional case, respectively. This separation is convenient for a clear presentation, and also reflects clearly the main differences between the modules associated to each of these superalgebras. The chapter finishes with detailed character formulae for the different types considered.
The final chapter corresponds to an active research subject, namely the super duality. The main objective of the authors is to obtain a complete solution to the irreducible character problem for the general linear and orthosymplectic superalgebras in certain BGG parabolic categories. By far the more demanding chapter, important questions like equivalence of categories, Kostant homology groups and Kazhdan-Lusztig-Vogan polynomials are inspected.
The appendices present the main properties of symmetric and supersymmetric functions, the Frobenius characteristic map as well as the boson-fermion correspondence and their relation to Schur functions.
At the end of each chapter, an interesting collection of exercises of varying difficulty is given, in order to develop particular questions in more detail.
Summarizing, this book not only constitutes a very good introduction to the subject of representation theory of Lie superalgebras, but also contains material of interest for the specialist, specifically concerning the super duality.

##### MSC:
 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 17B20 Simple, semisimple, reductive (super)algebras 17B22 Root systems 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)