Last update: 21 December 2013
Let be the Drinfel’d-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra Let us fix some notations. In particular, fix a triangular decomposition and let be the Weyl group of Let be the usual inner product on so that, if is a root, the corresponding reflection in is given by where The element is often viewed as an element of by using the form to identify and We shall use the conventions for quantum groups as in [Dri1990] and [LRa1977] so that and as algebras. The quantum group has a triangular decomposition corresponding to that of
If is a module and the weight space of is The category is the category of modules such that
|(b)||For all is finite,|
|(c)||is finitely generated as a module.|
If is a module generated by a highest weight vector of weight (i.e., a vector such that for and then any element of the center acts on by a constant, For each module let are the simple roots and Then where the sum is over all integrally dominant weights i.e., such that for all To summarize, there is a decomposition of the category where the sum is over all integrally dominant weights and is the full subcategory of modules such that The Grothendieck group of the category has bases where the integral Weyl group corresponding to is defines the dot action of on
Following the notations for the quantum group used in [LRa1977, §2], let and be the standard generators of the quantum group which satisfy the quantum Serre relations. The Cartan involution is the algebra anti-involution defined by The Cartan involution is a coalgebra homomomorphism. A contravariant form on a module is a symmetric bilinear form such that
Fix and such that is integrally dominant for all small positive real numbers Consider as an indeterminate and consider the Verma module as the module for generated by a vector such that for and There is a unique contravariant form such that Define The “specialization of at is and the Jantzen filtration of is By [Jan1980, Theorem 5.3], the Jantzen filtration is a filtration of by modules, the module is a maximal proper submodule of and each quotient has a nondegenerate contravariant form. It is known [Bar1983] that the Jantzen filtration does not depend on the choice of It is a deep theorem [BBe1993] that the quotients are semisimple and that if and are maximal length in their cosets and respectively, then the Kazhdan-Lusztig polynomial for is where is the length function on and is the multiplicity of the simple module in the factor of the Jantzen filtration of
Not all simple modules in the category have a BGG resolution. The general form of the BGG resolution given by Gabber and Joseph [GJo1981] is as follows.
Let be such that is dominant and regular and let be a parabolic subgroup of the integral Weyl group Let be the longest element of and fix Define a resolution of the simple module by Verma modules by setting where the sum is over all of length and defining the map where means that there is a (not necessarily simple) root such that and the maps are fixed choices of inclusions are fixed choices of signs such that Gabber and Joseph [GJo1981] prove that the sequence (2.7) is exact in this general setting. See [BGG1975] and [Dix1996, 7.8.14], for the original form of the BGG resolution. From the exactness of (2.7) it follows that if is dominant and regular then, in the Grothendieck group of the category where and is the longest element of
Let be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra There is an invertible element in (a suitable completion of) such that, for any two modules and the map is a module isomorphism. There is also a quantum Casimir element in the center of and, for a module we define In order to be consistent with the graphical calculus these operators should be written on the right. The elements and satisfy relations (see [LRa1977, (2.1-2.12)]), which imply that, for modules and a module isomorphism The relations (2.9) and (2.10) together imply the braid relation If is a module generated by a highest weight vector of weight then, by [Dri1990, Prop. 3.2], Note that are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. If is a finite dimensional module then is a direct sum of the irreducible modules and where is the projection onto in From the relation (2.11) it follows that if are finite dimensional irreducible modules then acts on the isotypic component of the decomposition by the constant where are positive integers. Suppose that and are modules with contravariant forms and respectively. Since the Cartan involution is a coalgebra homomorphism the form on defined by for is also contravariant. If is the Cartan involution defined in (2.4) then a formula of Drinfeld [Dri1990, Prop. 4.2], states from which it follows that Thus
This is a typed version of the paper Affine Braids, Markov Traces and the Category by Rosa Orellana and Arun Ram*.
*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.
This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.