Localization of Modules over Small Quantum Groups
نویسندگان
چکیده
We fix a base field k of characteristic not dividing l. We suppose that k contains a primitive l-th root of unity ζ , and fix it. Starting from these data, one defines certain category C. Its objects are finite dimensional X-graded k-vector spaces equipped with an action of Lusztig’s ”small” quantum group u (cf. [L2]) such that the action of its Cartan subalgebra is compatible with the X-grading. Variant: one defines certain algebra • u which is an ”X-graded” version of u (see 12.2), and an object of C is a finite dimensional • u-module. For the precise definition of C, see 2.11, 2.13. For l prime and chark = 0, the category C was studied in [AJS], for chark > 0 and arbitrary l, C was studied in [AW]. The category C admits a remarkable structure of a ribbon category (Lusztig).
منابع مشابه
Another Realization of the Category of Modules over the Small Quantum Group Sergey Arkhipov and Dennis Gaitsgory
0.1. Let g be a semi-simple Lie algebra. Given a root of unity (cf. Sect. 1.2), one can consider two remarkable Hopf algebras, Ul and ul, called the big and the small quantum group, respectively. Let Ul -mod and ul -mod denote the corresponding categories of modules. It is explained in [14] and [1] that the former is an analog in characteristic 0 of the category of algebraic representations of ...
متن کامل2 2 N ov 2 00 5 Representations of the Derivation Algebra of the Localization of the Quantum Plane at q = − 1 ( to appear in Comm . Alg . )
We determine the irreducible weight modules with weight multiplicities at most 1 over the derivation algebra of the localization of the quantum plane at q = −1.
متن کاملar X iv : m at h / 06 01 47 2 v 1 [ m at h . Q A ] 1 9 Ja n 20 06 REPRESENTATIONS OF QUANTUM GROUPS DEFINED OVER COMMUTATIVE RINGS II
In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.
متن کاملCrossed squares, crossed modules over groupoids and cat$^{bf {1-2}}-$groupoids
The aim of this paper is to introduce the notion of cat$^{bf {1}}-$groupoids which are the groupoid version of cat$^{bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}-$groupoids, and then we show their categories are equivalent....
متن کاملA Generalization of M-Small Modules
In this paper we introduce a generalization of M-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . We will use the structure of the radical of a module in and get some suitable results about this class of modules. Also the relation between injective hull in and this kind of modules will be investigated in this article. For a module we show...
متن کامل