Quiver varieties and t-analogs of q-characters of quantum affine algebras

Details Quiver varieties and t-analogs of q-characters of quantum affine algebras Quiver varieties and t-analogs of q-characters of quantum affine algebras

2001-05-22

arxiv.org/abs/math/0105173v2

Mathematics

Quantum Algebra

32 pages, The definition of the multiplication of the $t$--analog of the representation ring is corrected. Several ref's are a

Scientific paper

Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' $q$-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' will be explained in the end of the introduction.)

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