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**Infinite dimensional Lie algebras. 2nd ed.**
*(English)*
Zbl 0574.17010

Cambridge etc.: Cambridge University Press. XVII, 280 p. £25.00; $ 24.95 (1985).

The publication of a second edition just two years after the first one (BirkhĂ¤user 1983, Progr. Math. 44) proves the author’s presentation to be very successful and also shows the rapid development of this theory. For the contents of this monograph see B. Weisfeiler’s extensive review in Zbl 0537.17001.

Preface to the second edition. ”The most important additions reflect recent developments in the theory of infinite-dimensional groups (some key facts, like Proposition 3.8 and Exercise 5.19 are among them) and in the soliton theory (like Exercises 14.37-14.40 which uncover the role of the Virasoro algebra). The most important correction concerns the proof of the complete reducibility Proposition 9.10. The previous proof used Lemma 9.10 b) of the first edition which is false, as Exercise 9.15 shows. A correct version of Lemma 9.10 b) is the new Proposition 10.4 which gives a characterization of integrable highest weight modules.

In addition to correcting misprints and errors and adding a few dozen of new exercises, I have brought to date the list of references and related bibliographical comments.”

Preface to the second edition. ”The most important additions reflect recent developments in the theory of infinite-dimensional groups (some key facts, like Proposition 3.8 and Exercise 5.19 are among them) and in the soliton theory (like Exercises 14.37-14.40 which uncover the role of the Virasoro algebra). The most important correction concerns the proof of the complete reducibility Proposition 9.10. The previous proof used Lemma 9.10 b) of the first edition which is false, as Exercise 9.15 shows. A correct version of Lemma 9.10 b) is the new Proposition 10.4 which gives a characterization of integrable highest weight modules.

In addition to correcting misprints and errors and adding a few dozen of new exercises, I have brought to date the list of references and related bibliographical comments.”

Reviewer: O.Ninnemann

### MSC:

17B65 | Infinite-dimensional Lie (super)algebras |

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

05A19 | Combinatorial identities, bijective combinatorics |

11P81 | Elementary theory of partitions |

11F11 | Holomorphic modular forms of integral weight |

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |