Jan Schröer
FDAtlas
There are many different classes of algebras.
My aim is to write 23 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 finitedimensional, or at least there
will be a clear link with the world of finitedimensional 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 KacMoody Lie algebras, quantum groups, etc.
One of the core classes of finitedimensional 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.)

Quasitilted 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 quasitilted 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
(NonDynkin 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 finitedimensional 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 /
endowild / EndoWild / Corner endowild / Corner EndoWild.)

Homologically defined algebras


nNakayama 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 Skewedgentle algebras.)


More classes of algebras

Standard and nonstandard algebras 30.3.16
(These algebras are representationfinite.
Each representationfinite algebra is either standard or nonstandard.)

Minimal and Nagaseminimal algebras


Important classes of infinite dimensional algebras


Quantized enveloping algebras

KLR algebras (KLR = KhovanovLaudaRouquier)

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
More classes of algebras will be added.
You are welcome to send suggestions.