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Register a new userRingel-Hall algebras beyond their quantum groups I: Restriction functor and 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.
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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.
We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras.
In [19], Zheng studied the bounded derived categories of constructible $bar{mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the representations of a enlarged quiver and categorified the integrable highest weight modules of the corresponding quantum group by using these categories. In this paper, we shall generalize Zhengs work to highest weight modules of a subalgebra of the double Ringel-Hall algebra associated to a quiver in a functional version.
We give the Ringel-Hall algebra construction of the positive half of quantum Borcherds-Bozec algebras as the generic composition algebras of quivers with loops.
Let $Q$ be a finite acyclic valued quiver. We give the high-dimensional cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$.
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