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Register a new userThe Parity of Lusztigs Restriction Functor and Greens Formula
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Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringels Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Greens formula.
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In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in cite{Green} and the restriction functor in cite{Lusztig2}. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztigs framework.
In this paper, we give geometric realizations of Lusztigs symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztigs symmetries.
In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztigs perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztigs symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztigs symmetries on the whole quantum group.
Let $mathbf{U}$ be the quantized enveloping algebra and $dot{mathbf{U}}$ its modified form. Lusztig gives some symmetries on $mathbf{U}$ and $dot{mathbf{U}}$. Since the realization of $mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztigs symmetries on $mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $dot{mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztigs symmetries on $dot{mathbf{U}}$ by applying the BGP-reflection functors to $dot{mathcal{H}}$.
The geometric realizations of Lusztigs symmetries of symmetrizable quantum groups are given in this paper. This construction is a generalization of that in [19].
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