In Physics one is often interested in matrix solutions to certain systems of algebraic equations such as

Here the variables x and y are not supposed to commute with each other (since matrices usually do not commute). One problem is to find solutions that are truly different from each other. Certainly, pairs of square matrices of different sizes would be considered different. For matrices of a given size one has the notion of similarity which one knows from the theory of canonical forms. Note that if are -matrices that satisfy the system above and S is an a rbitrary invertible -matrix, then the pair

also is a solution. Such solutions will not be considered different.

The proper mathematical framework to accommodate the foregoing features is the representation theory of algebras over a field F (the ground field providing the entries for our matrices). In this theory one first defines an abstract object A in which the system of equations holds. In our case this would be the free algebra on the generators modulo the ideal I generated by the set

Thus has the requisite property with respect to the residue classes of x and y, which we will also denote by x and y. Now let M be a finite dimensional vector space over F and consider a homomorphism

from A into the F-algebra of linear transformations on M. Since respects addition and multiplication the transformations and also satisfy our sys tem of equations. By choosing a basis of M we produce a solution consisting of square matrices.

The map is called a representation of the associative algebra A, and M is referred to as an A-module. For ease of notation one writes

In this setting the similarity of matrices corresponds to the notion of an isomorphism between two A-modules M and N. Such an isomorphism is given by a bijective linear transformation such that

The main goal of the representation theory of algebras is the classification (up to isomorphisms) of the representations of an associative F-algebra A.

The theory of algebras and their representations is generally considered to originate in Hamilton's definition of Hypercomplex numbers and his famous construction of the quaternions. By the turn of the century algebras where successfully employed in the representation theory of finite groups. Maschke's Theorem states that the representations of the group algebras of finite groups over fields of characteristic zero are semisimple, that is that every representation is a direct sum of simple subrepresentations of which there are only finitely many. The structure of finite dimensional algebras with the latter property was completely determined by Wedderburn in 1908. His result was generalized by E. Artin in 1925 to algebras satisfying a somewhat weaker finiteness condition, the so-called Artin algebras. At around the same time the final version of the theorem of Krull-Remak-Schmidt was proved. This important result states that any finite dimensional A-module M can be written in an essentially unique way as a direct sum

of submodules, which in turn cannot be written as direct sums of proper submodules. This reduces the problem to the determination of these so-called indecomposable modules.

If the algebra A satisfies Maschke's Theorem it is called semisimple. In this simplest case, all indecomposable modules are simple, so that there are only finitely many indecomposable modules. There are other algebras with this property, and we say that an F-algebra A has finite representation type if it admits only finitely many non-isomorphic indecomposable modules. For instance, the theorem of finitely generated modules over principal ideal domains implies that the truncated polynomial algebra has finite representation type.

Unfortunately, most algebras are not of finite representation type. An early result by Higman asserts that a group algebra over a field of positive characteristic p>0 has finite representation type if and only if its p-Sylow subgroups are cyclic. If p=2 then our algebra above is the group algebra of the dihedral group of order 8. Thus, it is its own 2-Sylow subgroup, yet it is not cyclic, so it has infinite representation type.

According to a famous result due to Drozd, an algebra of infinite representation type is either tame or wild. In the former case all but finitely many indecomposable A-modules of a given dimension can be parametrized by essentially one parameter of the base field. It turns out that for p=2 the algebra defined above enjoys this property. If A is wild, then the classification of the indecomposables is harder than bringing two matrices simultaneously into Jordan form, a problem which is generally considered hopeless.

Since most algebras are of wild representation type, one has to find a new way of describing indecomposable modules. This is accomplished by the so-called Auslander-Reiten theory, which was initiated by M. Auslander and I. Reiten in the early seventies.