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**Growth of algebras and Gelfand-Kirillov dimension.**
*(English)*
Zbl 0564.16001

Research Notes in Mathematics, 116. Boston-London-Melbourne: Pitman Advanced Publishing Program. 182 p. £9.95 (1985).

Since its invention by I. M. Gelfand and A. A. Kirillov in 1966 [Publ. Math., Inst. Haut. Etud. Sci. 31, 5-20 (1966; Zbl 0144.021)], the notion of Gelfand-Kirillov (GK-)dimension has become an increasingly important tool in noncommutative ring theory, especially in the study of certain classes of algebras which are, in some sense, sufficiently closely related to commutative algebras. Prominent examples are enveloping algebras of Lie algebras, PI-algebras, and group algebras of nilpotent-by-finite groups. The first systematic study of the abstract aspects of GK-dimension was undertaken by W. Borho and H.-P. Kraft [in Math. Ann. 220, 1-24 (1976; Zbl 0306.17005)] which for a long time was the standard reference on GK-dimension. However, the exposition is marred by a number of inaccuracies and errors and, furthermore, the subject has witnessed a substantial increase of activity in the meantime. Thus an updated account was highly welcome.

The present book fully serves that purpose. It offers a reasonably complete survey of the state of the art, at least of its ring theoretic aspects. In the first half (Chapter 1-6), the definition and fundamental properties of GK-dimension of algebras, and of some related more general concepts, are laid out. Specifically, Chap. 1 is devoted to the notion of growth of algebras, a finer invariant than GK-dimension which was introduced by Borho and Kraft. Chapters 2-4 then study GK-dimension of algebras and its behavior under standard ring theoretic operations such as forming factor rings, skew polynomial rings, tensor products, and localizations. Chap. 5 describes Bernstein’s very useful extension of the notion of GK-dimension to modules, and in Chap. 6 graded and filtered algebras and modules and their GK-dimensions are considered. The second half of the book (Chapters 7-11) then applies these general techniques to special types of algebras. These include Duflo’s almost commutative algebras, where the classical Hilbert-Samuel polynomials from commutative dimension theory can be made to work (Chap. 7), Weyl algebras (Chap. 8), enveloping algebras of solvable Lie algebras (Chap. 9), PI-algebras (Chap. 10), and group algebras (Chap. 11). Amongst the results presented here, we mention only Makar-Limanov’s theorem on the existence of free subalgebras in the Weyl division algebra \(D_ 1\) and Gabber’s catenarity theorem for enveloping algebras of solvable Lie algebras. Often complete proofs are included, but occasionally the authors only fully describe an informative special case, or they concentrate on the part of the argument which depends on the use of GK-dimension. The results in this book have been drawn together from widely scattered sources, some had been unpublished before (e.g. Bergman’s theorem on algebras of GK-dimension \(<2)\), and the presentation is clear and very readable throughout.

The present book fully serves that purpose. It offers a reasonably complete survey of the state of the art, at least of its ring theoretic aspects. In the first half (Chapter 1-6), the definition and fundamental properties of GK-dimension of algebras, and of some related more general concepts, are laid out. Specifically, Chap. 1 is devoted to the notion of growth of algebras, a finer invariant than GK-dimension which was introduced by Borho and Kraft. Chapters 2-4 then study GK-dimension of algebras and its behavior under standard ring theoretic operations such as forming factor rings, skew polynomial rings, tensor products, and localizations. Chap. 5 describes Bernstein’s very useful extension of the notion of GK-dimension to modules, and in Chap. 6 graded and filtered algebras and modules and their GK-dimensions are considered. The second half of the book (Chapters 7-11) then applies these general techniques to special types of algebras. These include Duflo’s almost commutative algebras, where the classical Hilbert-Samuel polynomials from commutative dimension theory can be made to work (Chap. 7), Weyl algebras (Chap. 8), enveloping algebras of solvable Lie algebras (Chap. 9), PI-algebras (Chap. 10), and group algebras (Chap. 11). Amongst the results presented here, we mention only Makar-Limanov’s theorem on the existence of free subalgebras in the Weyl division algebra \(D_ 1\) and Gabber’s catenarity theorem for enveloping algebras of solvable Lie algebras. Often complete proofs are included, but occasionally the authors only fully describe an informative special case, or they concentrate on the part of the argument which depends on the use of GK-dimension. The results in this book have been drawn together from widely scattered sources, some had been unpublished before (e.g. Bergman’s theorem on algebras of GK-dimension \(<2)\), and the presentation is clear and very readable throughout.

Reviewer: M.Lorenz

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16Rxx | Rings with polynomial identity |

17B35 | Universal enveloping (super)algebras |