2012/03/07 by Brian Parshall, Parshall, Brian, Leonard L. Scott +1
Mathematics · #16G30 #16T20 #20G42 #81R50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1203.1550
openalex publication_date 2012/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \sO be a discrete valuation ring with fraction field K and residue field k. A quasi-hereditary algebra \wA over \sO provides a bridge between the representation theory of the quasi-hereditary algebra \wAK:=K⊗ \wA over the field K and the quasi-hereditary algebra Ak:=k⊗_\sO\wA over k. In one important example, \wAK--mod is a full subcategory of the category of modules for a quantum enveloping algebra while \wAk--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 \wA:= \bigoplusn≥ 0(\wA∩\radn\wAK)/(\wA∩\radn+1\wAK) is quasi-hereditary. A main result gives sufficient conditions that \gr\wA be quasi-hereditary. The main requirement is that each graded module \gr\wDelta(λ) arising from a \wA-standard (Weyl) module \wDelta(λ) have an irreducible head. An additional hypothesis requires that the graded algebra \gr \wAK be quasi-hereditary, a property recently proved by us to hold in some important cases involving quantum enveloping algebras. In the case where \wA 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\wA is quasi-hereditary. The proof depends on previous work of the authors, including a continuation of the methods there involving tightly graded subalgebras, and a development of a quantum deformation theory over \sO, worthy of attention in its own right, extending the work of Andersen-Jantzen-Soergel. As we point out, this work provides an essential step in our work on p-filtrations of Weyl modules for reductive algebraic groups over fields of positive characteristic.