A Category for the Adjoint Representation

By R. S. Huerfano, Mikhail Khovanov. 2000

97 citations recorded by the source corpus.

Abstract We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of C . The category C is the direct sum of a semisimple category and the category of modules over a certain algebra associated with a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root systems

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9bce93cd35d8af3a5b87ffa301d30c3a7c1d0d69

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Annular Khovanov homology and knotted Schur–Weyl representations (2015, background): “An analog of the theorem above for the (non super) current Lie algebra sl2(V(1)), where V(1) is the 2-dimensional irrep of sl2, is due to Loupias [26]; see also [18].”
A Geometric Model for the Module Category of a Gentle Algebra (2018, background): “Gentle algebras are Gorenstein [24], closed under tilting and derived equivalence [38, 39], and they are ubiquitous, for instance they occur in contexts such as Fukaya categories [27], dimer models [8], enveloping algebras of Lie algebras [29] and cluster theory.”
Braid cobordisms, triangulated categories, and flag varieties (2006, background): “• Actions generated by a chain of spherical objects [KS, ST, RZ, HK ].”
Hirzebruch–Riemann–Roch‐type formula for DG algebras (2007, unknown): “Namely, fix a non-zero element ω ∈ Λdim V V and set [26]:”
Quivers, Floer cohomology, and braid group actions (2000, result): “n in the derived category of modules over multiplicity one Brauer tree algebras, which are close relatives of Am(the idempotent (0) of Amis set to zero; these smaller algebras also appear in [38] and [22]). Our action essentially coincides with the one of Rouquier and Zimmermann. Acknowledgements. M”
A functor-valued invariant of tangles (2001, background): “It is particularly interesting to apply this construction to the action in example III of the a ne braid group B(Γ) in the derived category of A(Γ)-modules, for an a ne simply-laced Dynkin diagram Γ: The algebra A(Γ) is Morita equivalent to the cross-product (G) of the exterior algebra on 2 generato”