# Applications of spherical twist functors to Lie algebras associated to root categories of preprojective algebras

@inproceedings{Xu2022ApplicationsOS, title={Applications of spherical twist functors to Lie algebras associated to root categories of preprojective algebras}, author={Fan Xu and Fan Yang}, year={2022} }

Let ΛQ be the preprojective algebra of a finite acyclic quiver Q of non-Dynkin type and D(repΛQ) be the bounded derived category of finite dimensional nilpotent ΛQ-modules. We define spherical twist functors over the root category RΛQ of D (repΛQ) and then realize the Weyl group associated to Q as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra g(RΛQ) of RΛQ induced by spherical twist functors. We also present a conjectural relation between certain Lie subalgebras…

## References

SHOWING 1-10 OF 32 REFERENCES

Root Categories and Simple Lie Algebras

- Mathematics
- 1997

Abstract By using the T 2 -orbit category of the derived category of a hereditary algebra, which is proved to be a triangulated category too, we give a complete realization of a simple Lie algebra.…

Triangulated categories and Kac-Moody algebras

- Mathematics
- 2000

Abstract.By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T2=1, where T is the translation functor. In particular, the generic form of the…

Braid group actions on derived categories of coherent sheaves

- Mathematics
- 2000

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together…

Quivers, perverse sheaves, and quantized enveloping algebras

- Mathematics
- 1991

1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of…

Deriving DG categories

- Mathematics
- 1994

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5],…

Semicanonical Bases Arising From Enveloping Algebras

- Mathematics
- 2000

Abstract Let U+ be the plus part of the enveloping algebra of a Kac–Moody Lie algebra g with a symmetric Cartan datum. In [L1] we have defined a canonical basis of U+ under the assumption that the…

Quantum groups via Hall algebras of complexes

- Mathematics
- 2011

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z_2-graded complexes of quiver representations.