Special Lie algebras.

*(English. Russian original)*Zbl 0828.17006
Russ. Math. Surv. 48, No. 6, 111-152 (1993); translation from Usp. Mat. Nauk 48, No. 6(294), 103-140 (1993).

The existence of a polynomial identity for Lie algebras is not so strong a restriction on the algebras, as for associative algebras and many strong and beautiful results for associative algebras do not hold for Lie algebras with polynomial identity. It is a natural question to find a reasonably large class of Lie algebras which enjoy the nice properties of associative PI-algebras. Without any doubt such a class is the class of SPI (or special) Lie algebras, i.e. Lie algebras which have an associative enveloping algebra which satisfies a polynomial identity. The class of special Lie algebras was introduced by V. N. Latyshev in 1963 [Sib. Mat. Zh. 4, 1120-1121 (1963; Zbl 0128.258)]. Initially the interest to the SPI-algebras was related with the problem of describing Lie algebras whose universal enveloping algebra is PI, the problem if a homomorphic image of an SPI-algebra is also SPI and the finite basis problem for the polynomial identities for associative algebras. A survey on the early results is given in Yu. A. Bakhturin [Identities in Lie algebras, Nauka, Moscow (Russian) (1985; Zbl 0571.17001); English translation (Utrecht, VNU 1987)]. In the recent decade, a lot of new results have been obtained, mainly due to the efforts of the Moscow algebraic school, and now the theory of SPI-algebras is developed good enough. The paper under review is a well written survey on the topic and gives a good account on the results of the past decade.

Section 1 deals with the structure theory of SPI-algebras. It turns out that the finitely generated SPI-algebras enjoy a lot of properties both of finite-dimensional Lie algebras and finitely generated associative PI- algebras. Other topics discussed are the almost solvable Lie algebras, semisimple artinian and noetherian SPI-algebras, relatively free SPI- algebras.

Section 2 is devoted to the enveloping algebras of SPI-algebras – the existence of a polynomial identity for the universal enveloping algebra, the description of the PI-enveloping algebras of finite-dimensional algebras, the adjoint representation of SPI-algebras and the Grassmann hull of an SPI-algebra.

Section 3 studies the varieties of Lie algebras generated by SPI- algebras. The topics here are the polynomial identites of SPI-algebras and the finite-dimensional algebras and the finite basis problem, different ranks related with varieties of algebras and operations with special varieties.

Finally, Section 4 is devoted to the relations between the polynomial identities and the finiteness conditions – the Hopf property, ascending chain conditions on ideals, residual finiteness and presentation with finite-dimensional algebras over extensions of the base field.

The survey gives also some links with the theory of Lie superalgebras. A good list of references is included as well.

Section 1 deals with the structure theory of SPI-algebras. It turns out that the finitely generated SPI-algebras enjoy a lot of properties both of finite-dimensional Lie algebras and finitely generated associative PI- algebras. Other topics discussed are the almost solvable Lie algebras, semisimple artinian and noetherian SPI-algebras, relatively free SPI- algebras.

Section 2 is devoted to the enveloping algebras of SPI-algebras – the existence of a polynomial identity for the universal enveloping algebra, the description of the PI-enveloping algebras of finite-dimensional algebras, the adjoint representation of SPI-algebras and the Grassmann hull of an SPI-algebra.

Section 3 studies the varieties of Lie algebras generated by SPI- algebras. The topics here are the polynomial identites of SPI-algebras and the finite-dimensional algebras and the finite basis problem, different ranks related with varieties of algebras and operations with special varieties.

Finally, Section 4 is devoted to the relations between the polynomial identities and the finiteness conditions – the Hopf property, ascending chain conditions on ideals, residual finiteness and presentation with finite-dimensional algebras over extensions of the base field.

The survey gives also some links with the theory of Lie superalgebras. A good list of references is included as well.

Reviewer: V.Drensky (Sofia)

##### MSC:

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

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

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

17B05 | Structure theory for Lie algebras and superalgebras |

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