The ADE chain Workshop
Bielefeld, October 31 – November 1, 2008.
It will start on October 31, 13:15 and will end on November 1, around noon.
Room: V2-205 (second floor, part V of the building)
Note that there is a public holiday on Saturday, November 1. Thus, you will be allowed to enter the university that Saturday only through the main entrance. There, you will have to tell people from the security service about you attending the workshop. Please register such that we can give a complete list of participants to the university's security service.
Topic
There are several chains of algebras Rn with the following
property:
The derived category Db(Rn) is, for n = 1,2,3,…
of Lie type
A1, A2, A3,
D4, D5, E6,
E7, E8, C(2,3,5),
C(2,3,6), C(2,3,7), C(2,3,7)+, …
(where C(p,q,r) ist the canonical algebra of type (p,q,r), and where
C(p,q,r)+ is the extended canonical algebra (the one-point extension
using the simple projective module)).
Note that
- C(2,3,5) is derived equivalent to the extended Dynkin diagram to
E8, and
- C(2,3,6) is a tubular algebra.
For example:
- Take for Rn the path algebra of a linearly oriented quiver
of type An with arrows α modulo the relations α3 = 0.
- Or take for R2n the algebra T2(kAn),
where An is linearly oriented (and T2( – ) means to
form the upper triangular 2×2-matrix ring).
In case one deals with such a tensor product
one should look at the autormorphisms of the
derived category, which are given by the
Auslander-Reiten translations of the two factors,
namely
τA⊗1 and
1⊗τB. According to Takahashi,
such tensor decompositions are also of interest in
singularity theory.
This ADE-chain has been considered by many people:
- By Lenzing and de la Pena, looking at Coxeter polynomials
- By Ebeling, looking at singulaties
- By Happel and Seidel, looking at the algebras (1)
- By some esoteric physicists: see
D4-D5-E6-E7-E8 VoDou Physics Model
Better look at the references provided by A.D.King under
the heading "supergravity"!
-
- It is claimed by Lenzing that the derived category of each of
the algebras A2n or R2n is equivalent (as a
triangulated category) to the stable category of vector bundles
(modulo all line bundles) on the weighted projective line of type
(2,3,p) for p=(n+1)/2, or equivalently to the triangulated category
of graded singularities of the Brieskorn singularity
x2+y3+zp.
- Keller: Note that the cluster type of the homogeneous coordinate algebra
of the Grassmnannian G(3,n) is A_2, D_4, E_6, E_8, ... for
n=5, 6, 7, 8. See the survey of Fomin-Zelevinsky in
math.RT/0311493 Cluster algebras: Notes for the CDM-03 conference
- Keller: See also Barot-Geiss-Zelevinsky, p.4 of
math.CO/0411341 Cluster algebras of finite type and positive symmetrizable
matrices
- Keller: Note that T_2(kA_n) is the tensor product of kA_n and kA_2.
Anyone is invited to participate.
In case you are interested, please visit the registration page.
This is the webpage http://www.math.uni-bielefeld.de/~sek/sem/ADE-chain.html