2 5 A pr 2 00 9 The elements in crystal bases corresponding to exceptional modules ∗
نویسندگان
چکیده
According to the Ringel-Green Theorem([G],[R1]), the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum group([X],[XY]), in which the BGP-reflection functors coincide with Lusztig’s symmetries. We first assert the elements corresponding to exceptional modules lie in the integral generic composition algebra, hence in the integral form of the quantum group. Then we prove that these elements lie in the crystal basis up to a sign. Eventually we show that the sign can be removed by the geometric method. Our results hold for any type of Cartan datum.
منابع مشابه
2 00 9 The elements in crystal bases corresponding to exceptional modules ∗
According to the Ringel-Green Theorem([G],[R1]), the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, the Drinfeld double of the generic composition algebra can be identified with the whole quantum group([X],[XY]), in which the BGP-reflection functors coincide with Lusztig’s symmetries. We first assert the elements co...
متن کاملCrystal Bases and Monomials for Uq(G2)-modules
In this paper, we give a new realization of crystal bases for irreducible highest weight modules over Uq(G2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Introduction In 1985, the quantum groups Uq(g), which may be thought of as q-deformations of the universal enveloping algebras U(g) of Kac-Moody algebras g, were introd...
متن کاملA pr 2 00 8 REPRESENTATIONS OF POINTED HOPF ALGEBRAS AND THEIR DRINFEL ’ D QUANTUM DOUBLES
We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel’d quantum doubles. We construct a family of Vermatype modules in both cases and prove a parametrization theorem for the simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.
متن کاملG-frames in Hilbert Modules Over Pro-C*-algebras
G-frames are natural generalizations of frames which provide more choices on analyzing functions from frame expansion coefficients. First, they were defined in Hilbert spaces and then generalized on C*-Hilbert modules. In this paper, we first generalize the concept of g-frames to Hilbert modules over pro-C*-algebras. Then, we introduce the g-frame operators in such spaces and show that they sha...
متن کاملar X iv : 0 90 4 . 23 13 v 1 [ m at h . FA ] 1 5 A pr 2 00 9 A DISCRETIZED APPROACH TO W . T . GOWERS ’ GAME
We give an alternative proof of W.T. Gowers' theorem on block bases in Banach spaces by reducing it to a discrete analogue on specific count-able nets.
متن کامل