##
**Classification and identification of Lie algebras.**
*(English)*
Zbl 1331.17001

CRM Monograph Series 33. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4355-0/hbk). xi, 306 p. (2014).

The book is devoted to complete classification of Lie algebras mostly of low dimensions.

The well known Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of a semisimple Lie algebra and the maximal solvable ideal. While semisimple Lie algebras are completely classified the problem of complete classification of other Lie algebras seems to be unsolvable. In the book some class of nilpotent and solvable Lie algebras is chosen (this class contains algebras of arbitrary dimension) and their classification within this class is given. Also tables of indecomposable Lie algebra of dimension less than \(7\) are presented.

The book consists of four parts. The Part one is named “General theory”, it contains basic definition, formulation of the Levi theorem, the Cartan’s classification of semisimple algebras. The definition of generalized Casimir invariants is given and methods of its calculation are discussed (these invariants are used below in classification). The Part two is named “Recognition of a Lie algebra given by its structure constants”. In this part an algorithms are presented that allow to find out whether the algebra can be decomposed into direct sum, whether the algebra is solvable and an algorithm that allows to find the radical of a given Lie algebra and the Levi factor. The Part three is named “Nilpotent, solvable and Levi decomposed Lie algebras”. This part is devoted to classification of extensions of nilpotent Lie algebras to solvable ones. While the problem of complete classification of solvable Lie algebras seems to be unsolvable, there exist a classification of low-dimensional nilpotent Lie algebras. This strategy allows to perform some sort of classification of solvable Lie algebras. The Part four is named “Low-dimensional Lie algebras”, it consists of tables of all indecomposable Lie algebras of dimension \(n\), where \(1\leq n\leq 6\). The author considers his book as a tool for practitioners of Lie algebra theory, it is not intended to be a textbook, also it is not oriented to a specific application.

The well known Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of a semisimple Lie algebra and the maximal solvable ideal. While semisimple Lie algebras are completely classified the problem of complete classification of other Lie algebras seems to be unsolvable. In the book some class of nilpotent and solvable Lie algebras is chosen (this class contains algebras of arbitrary dimension) and their classification within this class is given. Also tables of indecomposable Lie algebra of dimension less than \(7\) are presented.

The book consists of four parts. The Part one is named “General theory”, it contains basic definition, formulation of the Levi theorem, the Cartan’s classification of semisimple algebras. The definition of generalized Casimir invariants is given and methods of its calculation are discussed (these invariants are used below in classification). The Part two is named “Recognition of a Lie algebra given by its structure constants”. In this part an algorithms are presented that allow to find out whether the algebra can be decomposed into direct sum, whether the algebra is solvable and an algorithm that allows to find the radical of a given Lie algebra and the Levi factor. The Part three is named “Nilpotent, solvable and Levi decomposed Lie algebras”. This part is devoted to classification of extensions of nilpotent Lie algebras to solvable ones. While the problem of complete classification of solvable Lie algebras seems to be unsolvable, there exist a classification of low-dimensional nilpotent Lie algebras. This strategy allows to perform some sort of classification of solvable Lie algebras. The Part four is named “Low-dimensional Lie algebras”, it consists of tables of all indecomposable Lie algebras of dimension \(n\), where \(1\leq n\leq 6\). The author considers his book as a tool for practitioners of Lie algebra theory, it is not intended to be a textbook, also it is not oriented to a specific application.

Reviewer: Dmitry Artamonov (Moskva)

### MSC:

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

17B05 | Structure theory for Lie algebras and superalgebras |

17B20 | Simple, semisimple, reductive (super)algebras |

17B30 | Solvable, nilpotent (super)algebras |

17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |