The publication of Dixmier’s book [Dix74] in 1974 led to increased interest
in the structure of enveloping algebras. Considerable progress was made in
both the solvable and semisimple cases. For example the primitive ideals
were completely classified and much information was obtained about the
structure of the primitive factor rings [BGR73], [Dix96], [Jan79], [Jan83],
Most of this work was complete by the early 1980s, so it was natural that
attention should turn to related algebraic objects. Indeed at about this time
some new noncommutative algebras appeared in the work of the Leningrad
school led by L. D. Faddeev on quantum integrable systems. The term
“quantum group” was used by V. G. Drinfel d and M. Jimbo to describe
particular classes of Hopf algebra that emerged in this way. This subject
underwent a rapid development, spurred on in part by connections with Lie
theory, low dimensional topology, special functions, and so on. The alge-
braic aspects of quantum groups are treated in detail in the books [CP95],
[Kas95], [KS97], [Lus93], [Jos95], and [Maj95].
Against this background, Lie superalgebras seem to have been somewhat
overlooked. Finite dimensional simple Lie superalgebras over algebraically
closed fields of characteristic zero were classified by V. G. Kac in his seminal
paper [Kac77a]. However more than thirty years after the classification, the
representation theory of these algebras is still not completely understood
and the structure of the enveloping algebras of these superalgebras remains
rather mysterious.
Previous Page Next Page