• Corpus ID: 247218381

# 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}
}```
• Published 19 January 2022
• Mathematics
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
Semicanonical bases and preprojective algebras II: A multiplication formula
• Mathematics
Compositio Mathematica
• 2007
Let \$\mathfrak{n}\$ be a maximal nilpotent subalgebra of a complex symmetric Kac–Moody Lie algebra. Lusztig has introduced a basis of \$U(\mathfrak{n})\$ called the semicanonical basis, whose elements
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
The Ringel–Hall Lie algebra of a spherical object
• Mathematics
J. Lond. Math. Soc.
• 2012
This work determines the Picard group of Sw and shows that each orbit category of Sw is triangulated and is triangle equivalent to a certain orbit categoryof the bounded derived category of a standard tube.
Quivers, perverse sheaves, and quantized enveloping algebras
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
— 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
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