A Note on Quantized Weyl Reciprocity at Roots of Unity
نویسندگان
چکیده
Let n, r be integers 1 and let V be a vector space of dimension n over C. Classically, we have natural commuting actions of the general linear group GL(V) and the symmetric group Sr on the tensor space V⊗r, and both actions satisfy the so-called Weyl reciprocity [W] (or the double centralizer property), which has been used by Schur and Weyl to link representations of Sr with representations of GL(V). This reciprocity is also true over any field by taking them up over Z in [CL]. That is, if AZ is the Kostant Z-form of the universal enveloping algebra A = A(gln(Q)) and V is Z-free and of rank n, then the natural homomorphisms
منابع مشابه
ar X iv : 0 81 0 . 16 21 v 1 [ m at h . Q A ] 9 O ct 2 00 8 DRINFEL ’ D DOUBLES AND SHAPOVALOV DETERMINANTS
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig’s small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument.
متن کامل9 Prime Spectra of Quantized Coordinate Rings
This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. The first part of the paper is expository. We survey a number of classes of quantized coordinate...
متن کاملWeyl Group Multiple Dirichlet Series I
Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet s...
متن کاملPrime Spectra of Quantized Coordinate Rings
This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. The rst part of the paper is expository. We survey a number of classes of quantized coordinate r...
متن کامل