Support varieties for quantum groups

Details Support varieties for quantum groups Support varieties for quantum groups

1997-11-12

arxiv.org/abs/q-alg/9711008v2

Mathematics

Quantum Algebra

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

Scientific paper

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 $U_{\xi}$ 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.

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