Jan Schröer

FD-Atlas



Chapters of Part 1 of the FD-Atlas:

  1. Representation-finite algebras
    directed, distributive, representation-finite (representation-infinite), standard (non-standard)
  2. Tame and wild algebras
    τ-tame, derived tame, generically tame, tame (n-domestic, domestic, linear growth, polynomial growth, exponential growth), wild (strictly wild, controlled wild, endo wild, controlled endo wild, WILD, strictly WILD)
  3. Hereditary algebras
    hereditary, path, preprojective, quasi-hereditary, Schur, semisimple, separable, species, strongly quasi-hereditary
  4. Tilted algebras
    τ-tilting finite, n-Auslander, n-CY tilted, n-hereditary, n-representation-finite, almost hereditary, Auslander, canonical, concealed, concealed canonical, fractionally CY, Jacobian, quasi-canonical, quasi-tilted, shod, tilted, tubular, twisted fractionally CY, weakly n-representation-finite (n-representation-infinite), weakly shod
  5. Selfinjective algebras
    Brauer graph, Brauer tree, Frobenius, group, Hopf, periodic, repetitive, selfinjective, symmetric, trivial extension, twisted periodic, weakly symmetric
  6. Gorenstein algebras
    n-Gorenstein, Iwanaga-Gorenstein, QF-3, weakly Gorenstein
  7. Biserial algebras
    biserial, clannish, gentle, Nakayama, skewed-gentle, special biserial, string
  8. Multiplicative basis algebras
    incidence locally hereditary, monomial, multiplicative basis (multiplicative Cartan basis, filtered multiplicative basis)
  9. Graded algebras
    differential graded, enveloping, Ginzburg dg, graded, Hochschild cohomology, Koszul, Koszul dual, quadratic, tensor, Yoneda
  10. Other algebras
    P-maximal, P-minimal, gendo-symmetric, local, low-dimensional, Nagase P-minimal, one-point extension, tree, triangular

The representation theory of finite-dimensional algebras is a relatively young area of mathematics. Its big bang or rather big bangs were Gabriel's classification of representation-finite quivers in 1970 and Auslander and Reiten's discovery of almost split sequences (aka Auslander-Reiten sequences) in 1975.

There is quite a large zoo of classes of finite-dimensional algebras which people study for various reasons. Many of these classes have a beautiful representation theory and often provide a link to other areas of mathematics or mathematical physics. Part 1 of the FD-Atlas is a compilation of short notes on the most important classes. (I identified about 100 of these up to now.) Usually, I will briefly define a class, give some examples, mention a few important results, and provide literature recommendations for further reading.

Part 2 contains a recollection of some fundamental results and techniques from the representation theory of finite-dimensional algebras. This includes an overview of the categories and subcategories which are frequently studied. I also give a list of general conjectures, e.g. the classical homological conjectures. Many more conjectures can be found in the various more specialized sections of Part 1.

There will be regular updates.