Generalized Weyl algebras and their representations. (English. Russian original) Zbl 0807.16027

St. Petersbg. Math. J. 4, No. 1, 71-92 (1993); translation from Algebra Anal. 4, No. 1, 75-97 (1992).
Summary: A class of algebras that generalizes Weyl algebras is introduced. For one variety of such algebras the simple (i.e. irreducible) modules are classified and the two-sided ideals are described. The Krull dimension and the global dimension are computed for certain classes of algebras. It is proved that the \(\text{Ext}^ n\)’s and \(\text{Tor}_ n\)’s of simple modules are finite-dimensional. Maximal commutative subalgebras are examined.


16S30 Universal enveloping algebras of Lie algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
17B35 Universal enveloping (super)algebras
16D25 Ideals in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras