Jan Schröer

FD-Atlas


There are many different classes of algebras. My aim is to write 2-3 pages for each of the classes mentioned below. Most of these classes have a beautiful representation theory and often provide a link to (or open up) an entire mathematical universe. Sometimes I won't mention the original articles and their authors, but rather give preference to a readable survey article which might be written by somebody else. I will update the very poor lists of references in the future. Most algebras will be finite-dimensional, or at least there will be a clear link with the world of finite-dimensional algebras.
Your are welcome to send comments and suggestions.
All this is influenced by my personal taste and ignorance.
Lacking a better idea I used the global dimension to structure the following list of algebras.
Global Dimension 0

  • Group algebras (characteristic does not divide the group order) 24.1.16
    (Semisimple. Still boring, but less boring than the general semisimple case.)

Global Dimension at most 1

  • Hereditary algebras and path algebras 5.2.16
    (Very exciting! Many links to Kac-Moody Lie algebras, quantum groups, etc. One of the core classes of finite-dimensional algebras.)

  • Tensor algebras of species

  • Weighted projective lines
    (See the notes on Canonical algebras below. The category of coherent sheaves on a weighted projective line forms an abelian, hereditary category with a tilting object. This is not a module category. Closely related to the representation theory of canonical algebras.)

Global Dimension at most 2

  • Tilted algebras 7.1.16
    (Just one "tilt" away from a hereditary algebra. Derived equivalent to a hereditary algebra.)

  • Concealed algebras 4.3.16
    (This is a special class of tilted algebras.)

  • Quasi-tilted algebras 4.3.16
    (Just one "tilt" away from a hereditary category. Derived equivalent to a hereditary category.)

  • Canonical algebras 4.3.16
    (This is a special class of quasi-tilted algebras. Includes notes on Weighted projective lines.)

  • Tubular algebras
    (Just a few "branch extensions" away from the path algebra of an extended Dynkin quiver.)

  • Auslander algebras 24.5.16
    (Auslander algebras have global dimension at most 2. Includes notes on Higher Auslander algebras, which have higher but still finite global dimension.)

  • Preprojective algebras
    (Non-Dynkin case: These are infinite dimensional.)

Finite Global Dimension

Infinite Global Dimension

  • Selfinjective / weakly symmetric / symmetric algebras (not semisimple) 3.3.16

  • Group algebras and blocks of group algebras (characteristic divides the group order)
    (See the notes on Group algebras above. All of these algebras are symmetric.)

  • Trivial extension algebras
    (See the notes on Repetitive algebras below.)

  • Brauer tree algebras and Brauer graph algebras

  • Preprojective algebras
    (Dynkin case: These are finite-dimensional selfinjective.)

Classes of algebras where the global dimension does not matter so much
Representation types

  • Wild algebras 4.4.16
    (wild / fully wild / Wild / Fully Wild / controlled wild / endo-wild / Endo-Wild / Corner endo-wild / Corner Endo-Wild.)

Homologically defined algebras

  • n-Nakayama algebras

  • Periodic algebras

Combinatorially defined algebras

  • Biserial algebras 30.3.16
    (Includes notes on Special biserial algebras, String algebras and Gentle algebras. All of these are biserial.)

  • Clannish algebras 30.3.16
    (Includes notes on Skewed-gentle algebras.)

More classes of algebras

  • Standard and non-standard algebras 30.3.16
    (These algebras are representation-finite. Each representation-finite algebra is either standard or non-standard.)

  • Minimal and Nagase-minimal algebras

Important classes of infinite dimensional algebras

  • Quantized enveloping algebras

  • KLR algebras (KLR = Khovanov-Lauda-Rouquier)

  • Cluster algebras
    (Introduced by Fomin and Zelevinsky in 2000. A class of (possibly infinitely generated) commutative algebras. Deep connections to many areas of mathematics and mathematical physics.)

Lie algebras

  • Kac-Moody Lie algebras
More classes of algebras will be added. You are welcome to send suggestions.