The n-Category Café

It might mean the mutation rules to change the transcendence basis (or “coordinate system”) which is part of the definition of cluster algebras.

I do not have a clear picture of the subject, but without doubt these mutation rules are related to cluster categories. Categorification in part of representation theory and also in knot theory (Khovanov homology etc.) is a bit different than what most of the people at this blog mean by categorification. Here people mean by categorification doing constructions up to isomorphisms and so on, to get categorical analogues. In representation theory, generators of some familiar algebras (braid groups, Coxeter groups, Temperley-Lieb algebra, quantum groups, Birman-Murakami-Wenzl algebra) are sometimes realized as functors at the level of some module categories or derived categories and so on. Some instances of such constructions they call categorification.

Now, many of the constructions mentioned invove combinatorics of quivers. Quivers are important in several situations, for example Dynkin quivers tell you about root systems, some quivers tell about the structure of derived categories of coherent sheaves on projective spaces
(Beilinson theorem from 1970-s: there are
now noncommutative versions, say a work of Hiroyuki Minamoto: also another version said in passing in a paper of Kontsevich and Rosenberg on nc smooth spaces and so on; one should also mention that the business of structure of derived category of coherent sheaves on projective spaces has much to do with some physics applications like ADHM construction of (multi)instantons in 1970-s and BGG-resolutions in CFT/vertex algebra business); quivers appear in representation theory
of associative algebras especially finite-dimensional ones and so on (key words: hereditary algebras, Auslander-Reiten quiver, tilting module etc.). Note that you have here two different worlds: some very free algebras like related to projective spaces, and some very small like finite-dimensional ones. Classical paper on combinatorics of Coxeter group and relation to the quiver comobinatorics (and path algebras) is

I.N. Bernstein, I.M. Gel’fand, V.A. Ponomarev: Coxeter functors and Gabriel’s theorem, Uspehi Mat. Nauk 28, no. 2 (170), 19-33, 1973 (engl. transl. Russ. Math. Surv. 28-2, 17-32, 1973).

Now, let us go back to quantum groups; they were realized partly in work of Lusztig and then also later Nakajima by finding generators as simple objects in some categories of perverse sheaves on some configuration spaces. Such realizations are useful as one can apply advanced tools of Bernstein, Gabber, MacPherson etc. from the theory of perverse sheaves to representation theory.
One of the discoveries there was the discovery of inetersting canonical and dual canonical bases in representations, tensor products of representations and so on. The business of various bases was active research for a long time and at some point people were interested in some special bases and their changes for the cells in quantum flag varieties. So somewhere there Zelevinski and Fomin found
some interesting changes between different bases with some positivity conditions. they axiomatized this in terms of classical and quantum cluster algebras. This appeared to be related to some previously unknown phenomena in algebraic geometry, related also to some problems in matrix theory sometimes called total positivity (when matrix has all entries positive and all minors are also positive). This got translated into sort of algebraic geometry with some new phenomena: for example so-called Laurent phenomenon.

At some point A. Buan, R. J. Marsh, B. Keller, I. Reiten, G. Todorov etc. found connections (easily found at arXiv) between Zelevinsky-Fomin cluster theory and tilting theory which is a central notion in the theory of representations of quivers and representations of hereditary algebras mentioned above. They found a notion of cluster category which is some quotient of the bounded derived category of the category of modules of the path algebra of a quiver, with good tilting properties.
B. Keller proved that cluster categories are themselves triangulated.
Work on “Calabi-Yau categories” is related to this. B. Keller also proved some conjecture related to them which enables a solution of certain integrable model in physics. I’ll post some references when I have time (though it’d be better somebody closer to that field gives more explanation and references).

math.RT/0412077
math.RT/0509198
math.RT/0402054
math.RT/0410187
Bernhard Keller’s lecture notes Cluster algebras and quiver representations can be found on his webpage.

S. Fomin, A. Zelevinsky, Cluster algebras I, Journal AMS 15 no2, 497-529, 2002
(and on arXiv)

A. A. Beilinson, Coherent sheaves on P^N
and problems of linear algebras, Funct. Anal. Appl. 12 (1978) 214-216

A. A. Beilinson,On th ederived category of perverse sheaves, in K-theory, arithmetic and geometry, LNM 1289 (Springer 1987)

D. Happel, I. Reiten, S. Smalo, Tilting in abelian categories and uqasitilted algebras, Mem. AMS 575 (1996)

A.I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Ak. Nauk SSSR, Ser mat 53 (1989) 25-44

D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, math.AG/0503632

For ADHM contruction see: Michael Atiyah, Vladimir G. Drinfel’d, Nigel. J. Hitchin, Yuri I. Manin, Phys. Lett. A65 (1978) 185-187

If somebody needs old 1990 survey of tilting theory by I. Assem (from Banach Center Publ. 26) let me know, I can send it in some form.