Forced Gradings in Integral Quasi-hereditary Algebras with Applications to Quantum Groups

摘要

Let O be a discrete valuation ring with fraction field K and residue field k. A quasi-hereditary algebra ? over O provides a bridge between the representation theory of the quasi-hereditary algebra ?_K:=K ? over the field K and the quasi-hereditary algebra A_k:= k ?? over K. In one important example, A_K-mod is a full subcategory of the category of modules for a quantum enveloping algebra while A_k-mod is a full subcategory of the category of modules for a reductive group in positive characteristic. This paper considers first the question of when the positively graded algebra gr ?:= ?_(n≥0)(?∩rad~n Ak)/(?∩rad~(n+1) Ak) is quasi-hereditary. A main result gives sufficient conditions that gr A be quasi-hereditary. The main requirement is that each graded module grΔ(λ) arising from a ?-standard (Weyl) module Δ(λ) have an irreducible head. An additional hypothesis requires that the graded algebra gr ?k be quasi-hereditary, a property recently proved [16] to hold in some important cases involving quantum enveloping algebras. In the case where ? arises from regular dominant weights for a quantum enveloping algebra at a primitive pth root of unity for a prime p > 2h - 2 (where h is the Coxeter number), a second main result shows that gr ? is quasi-hereditary. The proof of this difficult result involves interesting new methods involving integral quasi-hereditary algebras. It also depends on previous work [16] of the authors, including a continuation of the methods there involving tightly graded subalgebras, and a development of a quantum deformation theory over O = Z_((p))(ζ), where ζ is a pth root of unity. This (new) deformation theory, given in Appendix II (§7), extends work of Andersen-Jantzen-Soergel [2], and is worthy of attention in its own right. As we point out, the present paper provides an essential step in our work on p-filtrations of Weyl modules for reductive algebraic groups over fields of positive characteristic.