Representation of an associative algebra

of dimension

A homomorphism of the algebra over a field into the algebra of matrices , i.e. a mapping assigning to each a square matrix of order such that

(*)

where , . It is usually also required that the unit in corresponds to the identity matrix; sometimes is also required to be finite-dimensional.

Every indecomposable representation of a finite-dimensional semi-simple algebra is equivalent to a direct summand of the regular representation. Hence, every finite-dimensional semi-simple algebra is an algebra of finite (representation) type, i.e. has a finite number of non-isomorphic indecomposable representations. Non-semi-simple algebras can be both of finite and of infinite representation type (e.g. such is the algebra ). Algebras of infinite type are further divided into algebras of wild type, whose classification problem contains the unsolved problem on matrix pairs (i.e. the problem of simultaneously reducing to canonical form two linear operators on a finite-dimensional space), and algebras of tame type.

Basic problems studied in the representation theory of associative algebras are that of obtaining necessary and sufficient conditions for an algebra to belong to one of the types listed, as well as that of classifying the indecomposable representations in the finite and tame cases. In the general case these problems have not been solved. The description of algebras of finite or tame type and their representations has been obtained for algebras the square of whose radical equals zero (cf. [2], [4], [8]–[10]). The Brauer–Thrall problem has been solved, i.e. it has been proved that, over any field, an algebra of infinite type has indecomposable representations of arbitrary high dimension, while over a perfect field there are infinitely many dimensions in each of which there are infinitely many indecomposable representations (cf. [5], [7]). Any algebra of finite type over an algebraically closed field has a multiplicative basis, i.e. a basis for which the product of two arbitrary elements in it is either zero or belongs to the basis [6]. The problem of dividing the class of group algebras into tame and wild ones has been completely solved [1].

Strongly related with representations of associative algebras are representations of other objects: rings, partially ordered sets, lattices, boxes.

References

[1]	V.M. Bondarenko, Yu.A. Drozd, "Representation type of finite groups" J. Soviet Math. , 20 : 6 (1982) pp. 2515–2528 Zap. Nauchn. Sem. Leningr. Univ. , 71 (1977) pp. 24–41
[2]	S.A. Kruglyak, "Representations of algebras the square of whose radical equals zero" J. Soviet Math. , 3 : 5 (1975) pp. 629–636 Zap. Nauchn. Sem. Leningr. Univ. , 28 (1972) pp. 60–69
[3]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[4]	L.A. Nazarova, "Representations of quivers of infinite type" Math. USSR Izv. , 7 : 4 (1973) pp. 749–792 Izv. Akad. Nauk SSSR Ser. Mat. , 37 : 4 (1973) pp. 752–791
[5]	U. Fischbacher, "Une nouvelle preuve d'un théorème de Nazarova et Roiter" C.R. Acad. Sci. Paris , 300 (1984) pp. 259–263
[6]	R. Bautista, P. Gabriel, A. Roiter, L. Salmeron, "Representation finite algebras and multiplicative bases" Invent. Math. , 81 (1985) pp. 217–285
[7]	A.V. Roiter, "Unbounded dimensionality of indecomposable representations of an algebra with an infinite number of indecomposable representations" Math. USSR Izv. , 2 : 6 (1968) pp. 1223–1230 Izv. Akad. Nauk SSSR Ser. Mat. , 32 : 6 (1968) pp. 1275–1282
[8]	V. Dlab, C. Ringel, "Indecomposable representations of graphs and algebras" , Amer. Math. Soc. (1976)
[9]	P. Donovan, M.R. Freislich, "The representation theory of finite graphs and associated algebras" , Carleton Univ. (1974)
[10]	P. Gabriel, "Unzerlegbare Darstellungen I" Manuscripta Math. , 6 : 1 (1972) pp. 71–103

Comments

Thus, for an associative algebra over the field (with ), a representation of is an algebra homomorphism , where is a vector space over and denotes the -algebra of all (linear) endomorphisms of . A subrepresentation of the representation is given by a subspace of which is -invariant for all , and, in this case, one obtains a representation on , called a quotient representation. Given a representation of , there is the dual (or contragredient) representation of the opposite algebra of (this is the algebra on the underlying vector space of with multiplication defined by ); by definition, for , , . Cf. also Contragredient representation.

Let be a representation; instead of for , , one often writes just ; in this way becomes a left -module, and any left -module is obtained in this way. Given two representations and , a mapping from to is a linear transformation satisfying for , , or, rewritten, ; thus it is an -module homomorphism. If is a family of representations, their direct sum is the representation , where is the direct sum of vector spaces and , for all . The category of all representations of , equivalently, the category of (left) -modules, is an Abelian category. Note that if is a central idempotent of (i.e. and for all ) and is an -module, then and are -modules, is the direct sum of and , and . On the other hand, , where , , and one may regard as an -module and as an -module. Thus, dealing with representations of one may assume that is connected (that is, the only central idempotents of are and ).

A representation of is said to be simple (or irreducible) provided it is non-zero and the only proper subrepresentation is the zero representation. The Schur lemma asserts that the endomorphism ring of a simple representation is a division ring (cf. Ring with division). A representation of is said to be of finite length if there is a sequence of subrepresentations such that is simple, for ; such a sequence is called a composition series of , is its length, and the factors are called the composition factors (cf. also Composition sequence). If a representation has a composition series, then any two composition series have the same length, and there is a bijection between the composition factors of the two series (the Jordan–Hölder theorem). This may be formulated also as follows: the Grothendieck group of all finite-length representations modulo exact sequences is the free Abelian group on the set of isomorphism classes of simple representations. A representation of is called semi-simple if it is a direct sum of simple representations, or, equivalently, if any subrepresentation is a direct summand.

A representation of is said to be indecomposable if it cannot be written as the direct sum of two non-zero representations. If is an indecomposable representation of of finite length, then its endomorphism ring is a local ring. For a finite direct sum of representations with local endomorphism rings, all direct sum decompositions into indecomposable representations are equivalent (the Krull–Schmidt theorem, cf. Krull–Remak–Schmidt theorem). It follows that the Grothendieck group of all finite-length modules modulo split exact sequences is the free Abelian group on the set of isomorphism classes of indecomposable representations.

The algebra is said to be representation-finite if there are only finitely many isomorphism classes of indecomposable representations of ; it is called tame if it is not representation-finite but all families of indecomposable representations are -parametric, and wild if the category -mod of all finite-dimensional -modules involves the classification problem for pairs of square matrices up to simultaneous equivalence [a7]. Let be a finite-dimensional algebra. If the dimension of each finite-dimensional indecomposable representation is bounded, then is representation-finite (the first Brauer–Thrall conjecture, solved by V.A. Roiter [7]) and any representation is the direct sum of finite-dimensional indecomposable ones [a12]. The second Brauer–Thrall conjecture asserts that if is not representation-finite and is an infinite field, then there are infinitely many isomorphism classes of dimension , for infinitely many . The conjecture has been solved for perfect by R. Bautista [a3] and K. Bongartz [a5], see also [a11]. If is not representation-finite, then is either tame or wild and not both (Drozd's theorem [a7]). Certain minimal representation-infinite algebras have been classified by D. Happel and D. Vossieck [a9], and questions concerning minimal representation-infinite algebras over algebraically closed fields can be transferred to this list; in particular, one gets in this way a criterion for finite-representation type [a4], [a8]. In general, questions concerning finite-dimensional algebras over algebraically closed fields are treated by considering quivers with relations (see Quiver).

Let be a finite-dimensional algebra. If has no non-zero nilpotent ideal, then is said to be semi-simple. The algebra is semi-simple if and only if any representation of is semi-simple; in this case, the simple representations are just the indecomposable summands of the regular representation of . In general, let be the radical of (cf. Radical of rings and algebras), it is the maximal nilpotent ideal of and is semi-simple. The simple representations of are the indecomposable summands of ; up to isomorphism, there are only finitely many. The indecomposable projective representations are the direct summands of the regular representation of , the indecomposable injective representations are the duals of the regular representation of . Any indecomposable projective representation of has a unique simple quotient representation, any indecomposable injective representation of has a unique simple subrepresentation; in this way one obtains a bijection between the isomorphism classes of the simple -modules and the indecomposable projective -modules, as well as the indecomposable injective -modules.

The basic notions of modern representation theory are due to M. Auslander and I. Reiten [a1]: Given any indecomposable -module , there is a mapping which is minimal right almost split: it is not a split epimorphism, given any mapping which is not a split endomorphism, there is a with , and given with , then is an automorphism. If is projective, take for its maximal submodule and for the inclusion mapping. For non-projective, the minimal right almost split mapping is surjective, its kernel is indecomposable (and not injective), and the inclusion mapping is minimal left almost split (defined by the dual properties); also, any indecomposable non-injective -module occurs in this way as . These exact sequences with minimal left almost split and minimal right almost split are called almost-split sequences (or Auslander–Reiten sequences). They are uniquely determined by and by ; given , the corresponding -module can be calculated as follows: Take a minimal projective representation of , let , then ; the construction is called the Auslander–Reiten translation.

The Auslander–Reiten quiver of has as vertices the isomorphism classes of the finite-dimensional indecomposable -modules , and there is an arrow provided there exists an irreducible mapping (note that a mapping with indecomposable is called irreducible if is not invertible and given a factorization of , then is a split monomorphism or is a split epimorphism); in addition, is equipped with the Auslander–Reiten translation . The meshes of the Auslander–Reiten quiver are as follows: Given an indecomposable non-projective representation and an indecomposable representation , there is an irreducible mapping if and only if there is an irreducible mapping (this is the case if and only if is a direct summand of , where is the middle term of the almost-split sequence ). The Auslander–Reiten quiver of is an important combinatorial invariant of , often one may recover from . In case is connected and has a finite component, is representation-finite (Auslander's theorem, [a1]). Deleting from the vertices of the form with indecomposable injective, , and with indecomposable projective, , one obtains the stable Auslander–Reiten quiver . For representation-finite, the components of are related to Dynkin diagrams (cf. Dynkin diagram) [a13], [a10]. Using covering theory [a6], the study of representation-finite algebras can be reduced to that of representation-directed algebras (an algebra is called representation directed if there are only finitely many indecomposable representations and they can be ordered so that for ). The Auslander–Reiten quiver of a representation-directed algebra (and therefore the category -mod) can be constructed effectively [a14].

References

[a1]	M. Auslander, "Applications of morphisms determined by objects" R. Gordon (ed.) , Representation Theory of Algebras , M. Dekker (1978) pp. 245–327
[a2]	M. Auslander, I. Reiten, "Representation theory of Artin algebras III" Comm. in Algebra (1975) pp. 239–294
[a3]	R. Bautista, "On algebras of strongly unbounded representation type" Comment. Math. Helv. , 60 (1985) pp. 392–399
[a4]	K. Bongartz, "A criterion for finite representation type" Math. Ann. , 269 (1984) pp. 1–12
[a5]	K. Bongartz, "Indecomposables are standard" Comment. Math. Helv. , 60 (1985) pp. 400–410
[a6]	K. Bongartz, P. Gabriel, "Covering spaces in representation theory" Invent. Math. , 65 (1981) pp. 381–387
[a7]	Yu.A. Drozd, "Tame and wild matrix problems" V. Dlab (ed.) P. Gabriel (ed.) , Representation Theory II , Lect. notes in math. , 832 , Springer (1980) pp. 242–258
[a8]	P. Dräxler, "-Fasersummen in darstellungsendlichen Algebren" J. Algebra , 113 (1988) pp. 430–437
[a9]	D. Happel, D. Vossieck, "Minimal algebras of infinite representation type with preprojective component" Manuscripta Math. , 42 (1983) pp. 221–243
[a10]	D. Happel, U. Preiser, C.M. Ringel, "Vinberg's characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules" V. Dlab (ed.) P. Gabriel (ed.) , Representation Theory II , Lect. notes in math. , 832 , Springer (1980) pp. 280–294
[a11]	L.A. Nazarova, A.V. Roiter, "Categorical matrix problems and the Brauer–Thrall conjecture" , Kiev (1973) (In Russian)
[a12]	C.M. Ringel, H. Tachikawa, "QF-3 rings" J. Reine Angew. Math. , 272 (1975) pp. 49–72
[a13]	Chr. Riedtmann, "Algebren, Darstellungsköcher, Überlagerungen, und zurück" Comment. Math. Helv. , 55 (1980) pp. 199–224
[a14]	C.M. Ringel, "Tame algebras and integral quadratic forms" , Lect. notes in math. , 1099 , Springer (1984)
[a15]	D. Happel, "Triangulated categories in representation theory of finite dimensional algebras" , London Math. Soc. (1988)

C.M. Ringel