Infinite dimensional Lie superalgebras.

*(English)*Zbl 0762.17001
De Gruyter Expositions in Mathematics. 7. Berlin etc.: W. de Gruyter. x, 250 p. (1992).

The main concern of this book is colour Lie superalgebra. A Lie superalgebra is a \((\mathbb Z/2\mathbb Z)\)-graded Lie algebra; it arises in situations where some variables commute while others anticommute and has been described by M. Scheunert [The theory of Lie superalgebras. Lect. Notes Math. 716 (1979; Zbl 0407.17001)] and F. A. Berezin [Introduction to algebra and analysis with anticommuting variables (Russian) (Moscow 1983; Zbl 0527.15020); English translation (Reidel 1987; Zbl 0659.58001)].

To obtain a colour Lie superalgebra (cLs) \(L\) one takes a commutative coefficient ring \(K\) with group of units \(U\) containing 2 and 3 and a bilinear alternating form \(\varepsilon: G\times G\to U\), where \(G\) is an additive abelian group. Now \(L\) is a \(G\)-graded Lie \(F\)-algebra \(L=\oplus L_ g\) such that for any \(a\in L_ g\), \(b\in L_ h\), \[ [a,b]+\varepsilon(g,h)[b,a]=0,\quad [[a,b],c]=[a,[b,c]]- \varepsilon(g,h)[b,[a,c]]. \] Ordinary Lie and Lie superalgebras are included as special cases by taking \(G\) of order 1 or 2 (with suitable values for \(\varepsilon)\). A first step is to show that such algebras can be formed from an associative \(G\)-graded algebra by introducing an appropriate “colour” commutator. Much of the ordinary theory of Lie algebras can be taken over with little change, thus there is a universal enveloping algebra and a Poincaré-Birkhoff-Witt theorem. Free cLs’s can be constructed with a basis of regular monomials; as in the ordinary case, \(G\)-homogeneous subalgebras of free cLs’s are again free and the automorphism group is tame. The rest of the book studies identities in enveloping algebras, irreducible representations of Lie superalgebras, finiteness conditions of cLs’s with identities, and various composition techniques (such as free products with amalgamated subalgebra), explored by means of the diamond lemma.

Most of the topics discussed are of interest even for ordinary Lie algebras, and moreover have so far received little attention in the literature, except in the first-named author’s book [Identical relations in Lie algebras (VNU Science Press, Utrecht, 1987), translation from Moscow: Nauka (1985; Zbl 0571.17001)]. There is a clear-cut case for treating the wider class of Lie superalgebras, but the case for colour Lie superalgebras is less overwhelming (although we are assured that “there are a number of papers …where the authors show that the axioms below are not artificial.”). The text reads smoothly and lays a solid groundwork for a study of cLs’s. It is to be hoped that it will encourage further research in this topic which will lead on to the peaks.

To obtain a colour Lie superalgebra (cLs) \(L\) one takes a commutative coefficient ring \(K\) with group of units \(U\) containing 2 and 3 and a bilinear alternating form \(\varepsilon: G\times G\to U\), where \(G\) is an additive abelian group. Now \(L\) is a \(G\)-graded Lie \(F\)-algebra \(L=\oplus L_ g\) such that for any \(a\in L_ g\), \(b\in L_ h\), \[ [a,b]+\varepsilon(g,h)[b,a]=0,\quad [[a,b],c]=[a,[b,c]]- \varepsilon(g,h)[b,[a,c]]. \] Ordinary Lie and Lie superalgebras are included as special cases by taking \(G\) of order 1 or 2 (with suitable values for \(\varepsilon)\). A first step is to show that such algebras can be formed from an associative \(G\)-graded algebra by introducing an appropriate “colour” commutator. Much of the ordinary theory of Lie algebras can be taken over with little change, thus there is a universal enveloping algebra and a Poincaré-Birkhoff-Witt theorem. Free cLs’s can be constructed with a basis of regular monomials; as in the ordinary case, \(G\)-homogeneous subalgebras of free cLs’s are again free and the automorphism group is tame. The rest of the book studies identities in enveloping algebras, irreducible representations of Lie superalgebras, finiteness conditions of cLs’s with identities, and various composition techniques (such as free products with amalgamated subalgebra), explored by means of the diamond lemma.

Most of the topics discussed are of interest even for ordinary Lie algebras, and moreover have so far received little attention in the literature, except in the first-named author’s book [Identical relations in Lie algebras (VNU Science Press, Utrecht, 1987), translation from Moscow: Nauka (1985; Zbl 0571.17001)]. There is a clear-cut case for treating the wider class of Lie superalgebras, but the case for colour Lie superalgebras is less overwhelming (although we are assured that “there are a number of papers …where the authors show that the axioms below are not artificial.”). The text reads smoothly and lays a solid groundwork for a study of cLs’s. It is to be hoped that it will encourage further research in this topic which will lead on to the peaks.

Reviewer: Paul M. Cohn (London)

##### MSC:

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

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

17B01 | Identities, free Lie (super)algebras |

17B35 | Universal enveloping (super)algebras |

17B70 | Graded Lie (super)algebras |

17B75 | Color Lie (super)algebras |

16W50 | Graded rings and modules (associative rings and algebras) |

16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |

16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |