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Linear representations of partially ordered sets and vector space categories. (English) Zbl 0818.16009
Algebra, Logic and Applications. 4. Brooklyn, NY: Gordon and Breach Science Publishers. 499 p. (1992).
The study of representations of an arbitrary finite partially ordered set (poset) \(S\) over a field \(k\) was initiated by L. A. Nazarova and A. V. Rojter [in Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 5-31 (1972; Zbl 0336.16031)], who considered the problem of finding a canonical form for a certain set of matrices that are indexed by the elements of \(S\), under a specified set of admissible transformations that depend on the partial order in \(S\). The idea was to use representations of posets as a tool to attack the second Brauer- Thrall conjecture on representations of finite-dimensional algebras [R. Bautista, Comment. Math. Helv. 60, 392-399 (1985; Zbl 0584.16017)].
Roughly at the same time representations of several particular posets were studied by P. Gabriel [Manuscr. Math. 6, 71-103 (1972; Zbl 0232.08001)] in order to obtain a classification of quivers of finite representation type over \(k\), i.e., quivers with only a finite number of pairwise non-isomorphic indecomposable representations. When the field \(k\) is algebraically closed, representations of quivers can be identified with the representations of finite-dimensional basic hereditary \(k\)- algebras. The aforementioned classification yielded a complete description of finite-dimensional radical-square-zero \(k\)-algebras of finite representation type. Gabriel’s point of view on representations of posets was different from but closely related to that of Nazarova and Rojter: a representation of a finite poset \(S\) over an arbitrary field \(k\) is a finite-dimensional \(k\)-vector space \(V\) with a subspace \(V(s)\) associated to every element \(s\) of \(S\) such that the map sending \(s\) to \(V(s)\) is an (order-preserving) homomorphism of the poset \(S\) into the poset of subspaces of \(V\) ordered by inclusion [P. Gabriel, Symp. Math. 11, 81-104 (1973; Zbl 0276.16001)].
In the ensuing several years, the theory of representations of posets experienced a period of rapid development, which was largely fueled by major advances in the study of representations of finite-dimensional algebras, in particular, by the theory of representations of quivers and quivers with relations and by the discovery of almost split sequences made by M. Auslander and I. Reiten [Commun. Algebra 3, 269- 294 (1975; Zbl 0331.16027)]. It is therefore not surprising that the theory of representations of posets found numerous important applications to representations of finite-dimensional algebras, a closely related field that originally motivated the theory. Another important field of applications of the theory of representations of posets is the study of integral representations of orders.
Although a treatment of the theory of representations of posets can be found in books on representations of finite-dimensional algebras [P. Gabriel and A. V. Rojter, Representations of finite-dimensional algebras, Algebra VIII, Encycl. Math. Sci. 73, Springer (1992); C. M. Ringel, Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)], the book under review is the first that is exclusively devoted to and contains a comprehensive account of the theory. In particular, the author presents the classification of posets of finite representation type and of their indecomposable representations [M. Kleiner, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 32-41 (1972; Zbl 0345.06001) and ibid. 28, 42-54 (1972; Zbl 0345.06002)], the classification of posets of tame representation type [L. A. Nazarova, Izv. Akad. Nauk SSSR, Ser. Mat. 39, 963-991 (1975; Zbl 0362.06001)], and, without proofs, the classification of posets of polynomial growth and their representations [A. G. Zavadskij, Izv. Akad. Nauk SSSR, Ser. Mat. 55, 1007-1048 (1991; Zbl 0809.16011)]. The author also presents his own results on almost split sequences for representations of posets. Numerous examples illustrate the theory and indicate possible applications.
The book gives a fair account of the strengths and weaknesses of the theory of representations of posets at the present time. To the credit of the theory are several elegant classification theorems, results on almost split sequences and structure of the Auslander-Reiten quiver, and connections with the theory of integral quadratic forms. Among the weaknesses are a technical nature of many proofs, mostly relying on combinatorial arguments, and the use of several languages without a clear delineation of the roles played by each of them, which makes the subject fragmented.
The latter point deserves an elaboration. In addition to the aforementioned definitions of Nazarova-Rojter and Gabriel, other definitions of representations of posets were introduced by several mathematicians, for example, one due to R. Bautista and R. Martinez [in Proc. Conf. Ring Theory, Antwerp. 1978, Lect. Notes Pure Appl. Math. 51, 385-433 (1979; Zbl 0707.16003)], which was used to prove the existence of almost split sequences. To present his own results, the author of the book under review needs still another definition. He views the category of representations of a poset \(S\) as a certain full subcategory of the category of finitely generated modules over a \(k\)- algebra \(A\). Here \(A\) is the incidence algebra of the poset obtained from \(S\) by adjoining an element that is greater than all original elements of \(S\).
All these definitions are essentially equivalent, but each of them seems to be better suited than others for handling a particular problem. This is the way the subject has developed and has been presented in the book, where although the equivalence of different definitions is thoroughly discussed, the necessity of having so many languages is not addressed. It is natural to suggest that, perhaps, a really convenient definition has not yet been found.
The book contains a lot of presently available information on representations of partially ordered sets and thus is useful as a reference book. It does not seem suitable as an introduction to the subject because it lacks a unifying idea or point of view that will either carry the beginners through the technical difficulties, or will replace the current treatment with a conceptual exposition of the subject.

MSC:
16G20 Representations of quivers and partially ordered sets
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16D90 Module categories in associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G30 Representations of orders, lattices, algebras over commutative rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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