Skew PBW monads and representations

Abstract

Let R be an associative ring, θ a map from some set G to the group Aut(R) of automorphisms of the ring R such that the image of θ is a semigroup in Aut(R). A skew PBW (Poincaré-Birkhoff-Witt) ring related to the map θ is an associative ring R{θ,ξ} which contains R as a subring and is a free right R-module with a basis {x _s | s ∈ G} such that rx _s = θ _s (r)x _s for any s ∈ G and all r ∈ R. The symbol ξ stays for the multiplication table: x _s x _t = Σx _u ξ (s,t | u). We assume that G has a marked element * which the map θ sends into id _R ; and x _* = 1 — the identity element of the ring R. Important special cases of skew PBW rings are (quantized) enveloping algebra of an arbitrary Kac-Moody Lie algebra, enveloping algebras of reductive Lie algebras, Heisenberg and Weyl algebras (of an arbitrary rank) and their ‘quantum’ deformations, enveloping algebra of. the Virasoro Lie algebra, crossed products. The main result of this chapter, Theorem 6.6.3, describes (gives a canonical realization of) the spectrum of the category R{θ,ξ} — mod in terms of the spectrum of R — mod. Recall that annihilaters of modules from the spectrum are prime ideals, and if the ring is left noetherian, any prime ideal is the annihilator of a module of the spectrum (cf. Corollary I.6.4.6). Another fact is that the spectrum of a category contains isomorphy classes of all simple objects. And Theorem 6.6.3 allows to single out immediately a series of irreducible representations of R{θ,ξ} which could be called generalized Harish-Chandra modules. A more involved application of Theorem 6.6.3 (and some other facts of the developed in Chapters III, IV noncommutative spectral theory) allows to find some natural classes of non-diagonalizable irreducible representations of reductive Lie algebras and Kac-Moody Lie algebras.