##
**Enveloping algebras.**
*(English)*
Zbl 0867.17001

Graduate Studies in Mathematics. 11. Providence, RI: American Mathematical Society (AMS). xx, 379 p. (1996).

This is a new printing of the English edition of the extraordinary book by Jacques Dixmier devoted to the theory of universal enveloping algebras of finite dimensional Lie algebras over a field of characteristic zero. For reviews of earlier editions see Zbl 0308.17007 (Paris 1974), Zbl 0339.17007 (Amsterdam 1977), Zbl 0346.17010 (Berlin 1977).

Starting with a locally compact group \(G\), an important problem is to determine the continuous representations of \(G\) in Hilbert spaces or even in more general topological vector spaces. Let \(G\) be a real Lie group with a complex Lie algebra \(\mathfrak g\). The study of finite dimensional representations of \(G\) is almost equivalent to that of the finite dimensional representations of \(\mathfrak g\). Although for infinite dimensional representations the relationship is more delicate, it is reasonable to consider the representations of \(\mathfrak g\). Passing to the universal enveloping algebra \(U({\mathfrak g})\), the study of the representations of \(\mathfrak g\) is transformed into a similar problem for the associative algebra \(U({\mathfrak g})\). This approach has the disadvantage of working with the infinite dimensional algebra \(U({\mathfrak g})\) but allows to apply powerful associative methods as maximal left ideals, localization, etc. The kernel of an irreducible representation of \(U({\mathfrak g})\) is a primitive ideal. Even if the set of classes of non-equivalent irreducible representations of \(\mathfrak g\) is very large, the set of primitive ideals of \(U({\mathfrak g})\) can be of reasonable size. The leading idea of the book is to study primitive ideals of \(U({\mathfrak g})\) instead of irreducible representations of \(\mathfrak g\).

The purpose of the first five chapters is to introduce the principal tools for investigation of primitive ideals of enveloping algebras. The study of enveloping algebras requires a good knowledge of Lie algebras. On the other hand, some properties of Lie algebras can be established by techniques based on enveloping algebras (and the author exploits actively this opportunity). Chapter 1 contains the properties of Lie algebras needed in the rest of the book. Chapter 2 introduces the enveloping algebras and Chapter 3 gives information about their two-sided ideals. Chapter 4 is a link between enveloping and commutative algebras and deals with the centres of enveloping algebras, of their quotients and their rings of fractions. Chapter 5 is devoted to the concept of induced representations and its variants because they play an important role in the construction of irreducible representations of Lie algebras (and hence of primitive ideals of enveloping algebras).

In Chapter 6 the author uses the Dixmier-Kirillov orbital method from the theory of nilpotent Lie groups. He describes the primitive ideals of \(U({\mathfrak g})\) when \(\mathfrak g\) is solvable and the base field is algebraically closed. The next three chapters deal with semi-simple Lie algebras \(\mathfrak g\). Chapters 7 and 9 are devoted to some particular representations linked, respectively, with the choice of a Cartan subalgebra and with a symmetric decomposition. Chapter 8 gives the minimal primitive ideals of \(U({\mathfrak g})\) over an algebraically closed field. Chapter 10 is based on the results of Chapters 1-8. Using the orbital method in a suitable form, the author constructs a large family of primitive ideals of \(U({\mathfrak g})\). Chapter 11 is an appendix containing basic information about root systems and some miscellaneous results.

Each chapter closes with supplementary remarks specifying bibliographical questions, giving counterexamples and indicating additional results. The book finishes with a list of 40 open problems. For the new printing the author has updated the list of references and the current state of the problems.

The text of the book is self-contained, written with precision and elegance. Written more than 20 years ago, the book is still an excellent textbook for the graduate student, a very good background for the professional algebraist not very familiar with the subject and a very useful source of references for the expert.

Starting with a locally compact group \(G\), an important problem is to determine the continuous representations of \(G\) in Hilbert spaces or even in more general topological vector spaces. Let \(G\) be a real Lie group with a complex Lie algebra \(\mathfrak g\). The study of finite dimensional representations of \(G\) is almost equivalent to that of the finite dimensional representations of \(\mathfrak g\). Although for infinite dimensional representations the relationship is more delicate, it is reasonable to consider the representations of \(\mathfrak g\). Passing to the universal enveloping algebra \(U({\mathfrak g})\), the study of the representations of \(\mathfrak g\) is transformed into a similar problem for the associative algebra \(U({\mathfrak g})\). This approach has the disadvantage of working with the infinite dimensional algebra \(U({\mathfrak g})\) but allows to apply powerful associative methods as maximal left ideals, localization, etc. The kernel of an irreducible representation of \(U({\mathfrak g})\) is a primitive ideal. Even if the set of classes of non-equivalent irreducible representations of \(\mathfrak g\) is very large, the set of primitive ideals of \(U({\mathfrak g})\) can be of reasonable size. The leading idea of the book is to study primitive ideals of \(U({\mathfrak g})\) instead of irreducible representations of \(\mathfrak g\).

The purpose of the first five chapters is to introduce the principal tools for investigation of primitive ideals of enveloping algebras. The study of enveloping algebras requires a good knowledge of Lie algebras. On the other hand, some properties of Lie algebras can be established by techniques based on enveloping algebras (and the author exploits actively this opportunity). Chapter 1 contains the properties of Lie algebras needed in the rest of the book. Chapter 2 introduces the enveloping algebras and Chapter 3 gives information about their two-sided ideals. Chapter 4 is a link between enveloping and commutative algebras and deals with the centres of enveloping algebras, of their quotients and their rings of fractions. Chapter 5 is devoted to the concept of induced representations and its variants because they play an important role in the construction of irreducible representations of Lie algebras (and hence of primitive ideals of enveloping algebras).

In Chapter 6 the author uses the Dixmier-Kirillov orbital method from the theory of nilpotent Lie groups. He describes the primitive ideals of \(U({\mathfrak g})\) when \(\mathfrak g\) is solvable and the base field is algebraically closed. The next three chapters deal with semi-simple Lie algebras \(\mathfrak g\). Chapters 7 and 9 are devoted to some particular representations linked, respectively, with the choice of a Cartan subalgebra and with a symmetric decomposition. Chapter 8 gives the minimal primitive ideals of \(U({\mathfrak g})\) over an algebraically closed field. Chapter 10 is based on the results of Chapters 1-8. Using the orbital method in a suitable form, the author constructs a large family of primitive ideals of \(U({\mathfrak g})\). Chapter 11 is an appendix containing basic information about root systems and some miscellaneous results.

Each chapter closes with supplementary remarks specifying bibliographical questions, giving counterexamples and indicating additional results. The book finishes with a list of 40 open problems. For the new printing the author has updated the list of references and the current state of the problems.

The text of the book is self-contained, written with precision and elegance. Written more than 20 years ago, the book is still an excellent textbook for the graduate student, a very good background for the professional algebraist not very familiar with the subject and a very useful source of references for the expert.

Reviewer: V.Drensky (Sofia)

### MSC:

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

17B35 | Universal enveloping (super)algebras |

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

17B30 | Solvable, nilpotent (super)algebras |

17B45 | Lie algebras of linear algebraic groups |

16S30 | Universal enveloping algebras of Lie algebras |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |