Support varieties for quantum groups

1997/11/12 by V. Ostrik, Ostrik, V.
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #q-alg

paper · pdf · doi:10.48550/arxiv.q-alg/9711008

11 pages. The proof of the Theorem 4.5 is extended in order to avoid a gap found by J.C.Jantzen

arxiv created 1998/02/09 · arxiv updated 2009/11/30

Abstract

For any module M over small quantum group one defines the support variety using construction from the theory of restricted Lie algebras. It is a closed conical subset of nilpotent cone of the corresponding Lie algebra. If module M is a module over the quantum group U_ξ with divided powers then its support variety is invariant under the action of the corresponding algebraic group. In this case we relate codimension 2a of the support variety of M in nilpotent cone and dimension of M. Namely, we prove that dim M is `almost' divisible by l^a. Further, we give an a priori estimate for support variety of a module in a given linkage class. We compute the support varieties for Weyl modules. Also we compute the support varieties for tilting modules over quantum SL_n and verify in this case Humphreys' Conjecture which relates support varieties of tilting modules with Lusztig's bijection between nilpotent orbits and two-sided cells in the affine Weyl group.

Related

• Another realization of the category of modules over the small quantum group
• Conjectures on tilting modules and antispherical p-cells
• Varieties for Modules of Quantum Elementary Abelian Groups
• Quantum Groups, the loop Grassmannian, and the Springer resolution
• Cohomology for quantum groups via the geometry of the nullcone