Differential operators on quantized flag manifolds at roots of unity

2010/01/31 by Toshiyuki Tanisaki, Tanisaki, Toshiyuki
Mathematics · #17B37 #20G05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1002.0113

openalex publication_date 2010/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The quantized flag manifold, which is a q-analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of D-modules on it using the framework of non-commutative algebraic geometry; however, when the parameter q is a root of unity, Lusztig's Frobenius morphism allows us to handle D-modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold. In this paper we will show that this sheaf of rings on the ordinary flag manifold is an Azumaya algebra over its center. We also show that its restriction to certain subsets are split Azumaya algebras. These are analogues of some results of Bezrukavnikov-Mirković-Rumynin on D-modules on flag manifolds in positive characteristics.

Related

• Differential operators on quantized flag manifolds at roots of unity III
• Categories of D-modules on a quantized flag manifold
• The ring of differential operators on a quantized flag manifold
• Quantized flag manifolds and non-restricted modules over quantum groups\n at roots of unity
• Affine open covering of the quantized flag manifolds at roots of unity