Exceptional collections

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\underline{Exceptional collections}
 
The purpose of this talk is to see a class of triangulated categories
which can
be described in terms of a finite amount of explicit information. There is
a lot
of literature, but the papers tend to repeat each other a fair bit. The
prerequisites
are some algebra, in particular knowledge of quivers, and some algebraic
geometry 
(on the abstract side).
 
\underline{Plan}
 
{\it Exceptional collections, mutation.} Let's work in the framework of
classical
triangulated categories over a field ($\C$ is actually enough for us). We
need
the notion of full exceptional collection, and most importantly, that of
mutation
of such collections. A good survey article is [Gorodentsev-Kuleshov, helix
theory].
The most important thing is to see the braid group arising. 
 
Besides the abstract theory, we also want concrete examples, at least the
two basic 
exceptional collections on the projective plane (that consisting of line
bundles, and
its dual, which is Beilinson's collection). We should check that these are
indeed
obtained by mutation.
 
{\it The cochain level theory.} Here, we assume that our triangulated
category is
the derived category of an $A_\infty$-category. One can then associate to
each
exceptional collection a finite $A_\infty$-subcategory which is directed
(this
term should be defined, and it should be emphasized that such categories
only
have finitely many nonzero products). References are [Seidel, Fukaya
categories
and Picard-Lefschetz theory, Chapter 1] or [Kontsevich, ENS lecture notes
from
1998, never published but I have a copy].
 
If the collection is full, the directed $A_\infty$-category
is derived equivalent to the usual one. The theory of mutations now looks
differently,
namely we have a braid group action on the set of quasi-isomorphism
classes of
directed $A_\infty$-categories. 
 
A very interesting example comes from representation theory
[Bernstein-Gelfand-Ponomarev]: 
directed quiver algebras of type ADE have finite orbits under the braid
group action.
We do want to see the example of the A-chain with three objects.
 
\underline{Notes}
 
[Seidel, book] makes a big deal out of treating $A_\infty$-categories
which have
units only on the cohomology level. However, for our purposes here, it's
ok to assume
strict units, in which case the construction of the directed
$A_\infty$-subcategory
is much more obvious.
 
\underline{Dependencies}
 
This only relies on $A_\infty$-structure theory and the notion of derived
category of an $A_\infty$-category.
 
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