Representations Of Finite-dimensional Algebras Research Articles

Exact hierarchical reductions of dynamical models via linear transformations

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model’s formal analysis and mechanistic understanding. We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction. This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint. The algorithm reduces finding exact linear reductions to a question about representations of finite-dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely available at https://github.com/x3042/ExactODEReduction.jl.

Quantum Affine Algebras, Graded Limits and Flags

In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations of these algebras. This connection arises via the notion of the graded and classical limit of finite-dimensional representations of quantum affine algebras. We explain how this study has led to interesting connections with Macdonald polynomials and discuss a BGG-type reciprocity result. We also discuss the role of Demazure modules in this theory and several recent results on the presentation, structure and combinatorics of Demazure modules.

Infinite dimensional representations of finite dimensional algebras and amenability

We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

Complexity classes of modules over finite dimensional algebras

Directed graphs called “syzygy quivers” are used to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finite dimensional syzygy-finite representation (e.g. any representation of a monomial algebra), we show that this growth rate is poly-exponential, i.e. the product of a polynomial and an exponential function, and give a procedure for computing the corresponding degree and base from a syzygy quiver. We characterize the growth rates arising in this context. The bases of the occurring exponential functions are the Perron numbers: real, nonnegative algebraic integers b whose irreducible polynomial over Q has no root with modulus larger than b. Modulo this restriction, arbitrary degrees and bases occur. Moreover, we show that these growth rates are invariant under stable derived equivalences.

Embeddings of Representations

We derive “numerical” criteria for the existence of embeddings of representations of finite dimensional algebras.

Pullback Moduli Spaces

Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty which occurs in various natural cases is that nonisomorphic modules are sent to the same point in the moduli spaces which arise. In this article, we study how this collapsing phenomenon can sometimes be reduced by considering pullbacks of modules for an auxiliary algebra. One application is a geometric proof that the twisting action of an algebra automorphism induces an algebraic isomorphism between moduli spaces.

On the *-representation of one class of algebras associated with Coxeter graphs

We investigate *-representations of a class of algebras that are quotient algebras of the Hecke algebras associated with Coxeter graphs. A description of all unitarily nonequivalent irreducible *-representations of finite-dimensional algebras is given. We prove that only trees that have at most one edge of type s > 3 define algebras of finite Hilbert type for all values of parameters.

Root-induced integral quadratic forms

Given an integral quadratic unit form q : Z n → Z and a finite tuple of q-roots r = ( r j ) j∈ J the induced q-root form q r is considered as in [P. Gabriel, A.V. Roiter, Representations of finite dimensional algebras, in: A.I. Kostrikin, I.V. Shafarevich (Eds.), Algebra VIII, Encyclopaedia of the Mathematical Sciences, vol. 73, 1992, Springer (Chapter 6)]. We show that two non-negative unit forms are of the same Dynkin type precisely when they are root-induced one from the other. Moreover, there are only finitely many non-negative unit forms without double edges of a given Dynkin type. Root-induction yields an interesting partial order on the Dynkin types, which is studied in the paper.

The Category of Morphisms Between Projective Modules#

Let Λ be an artin algebra over a local commutative artinian ring k. We consider the category P(Λ) whose objects are morphims f : P → Q with P and Q projective left Λ-modules. In P(Λ), we introduce an exact structure in the sense of Gabriel and Roiter [Gabriel, P., Roiter, A. V. (1992). Representations of finite-dimensional algebras. In: Kostrikin, A. I., Shafarevich, I. V. eds. Encyclopaedia of the Mathematical Sciences. Vol. 73. Springer. Algebra VIII.] or equivalently Quillen [Quillen, D. (1973). Higher Algebraic K-Theory, 1. SLNM 341. Berlin: Springer, pp. 85–147], then we describe the corresponding projectives and injectives. For p(Λ), the full subcategory whose objects are morphisms f : P → Q with P and Q finitely generated, we prove a relation between the Hom-functor and the Ext-functor. From here, we can prove the existence of almost split sequences in p(Λ) and its relation with the almost split sequences for left Λ-modules.

Relative invariants for representations of finite dimensional algebras

Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d M β on the variety of modules of dimension vector β, such that d M β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d M β with M indecomposable.

Optimal filtrations on representations of finite dimensional algebras

We present a representation theoretic description of the non-empty strata in the Hesselink stratification of the nullcone of representations of quivers. We use this stratification to define optimal filtrations on representations of finite dimensional algebras. As an application we investigate the isomorphism problem for uniserial representations. 1. OPTIMAL FILTRATIONS 1.1. Let A be a finite dimensional algebra over an algebraically closed field k. In this paper we want to parameterize isomorphism classes of finite dimensional Amodules having a specific Jordan-Holder sequence. In particular, we want to relate the recent results of K. Bongartz and B. Huisgen-Zimmermann [3, 4, 1] on uniserial modules to the Hesselink stratification of nullcones. By Morita theory we may reduce to the case that A is a basic algebra. That is, all simple A-modules are one dimensional. In this case we can write A as the quotient of the path algebra of a quiver and relate finite dimensional A-modules to representations of this quiver. 1.2. A quiver Q is a directed graph on a finite set of vertices {vl,..., Vn}. Let aij be the number of directed arrows from vi to vj (or loops if v = vj). The Euler-form of Q is the bilinear form X: Z x Zn Z determined by the matrix X = (Xij)i,j c Mn(Z) with entries Xij = 6ij aij Clearly, X encodes the structure of the directed graph Q. A representation V of a quiver Q of dimension vector a = (al,..., an) E Nn assigns to every arrow vi -vj in Q a matrix V(Q) E Majixi (k). The set of all a-dimensional representations form an affine space,

The geometry of uniserial representations of finite dimensional algebras I

It is shown that, given any finite dimensional, split basic algebra λ = KΓ I (where Γ is a quiver and I an admissible ideal in the path algebra KΓ), there is a finite list of affine algebraic varieties, the points of which correspond in a natural fashion to the isomorphism types of uniserial left Λ-modules, and the geometry of which faithfully reflects the constraints met in constructing such modules. A constructive coordinatized access to these varieties is given, as well as to the accompanying natural surjections from the varieties onto families of uniserial modules with fixed composition series. The fibres of these maps are explored, one of the results being a simple algorithm to resolve the isomorphism problem for uniserial modules. Moreover, new invariants measuring the complexity of the uniserial representation theory are derived from the geometric viewpoint. Finally, it is proved that each affine algebraic variety arises as a variety of uniserial modules over a suitable finite dimensional algebra, in a setting where the points are in one-one correspondence with the isomorphism classes of uniserial modules.

Semicontinuity for representations of one-dimensional Cohen-Macaulay rings

We show that the number of parameters for CM-modules of prescribed rank is semi-continuous in families of CM rings of Krull dimension 1. This transfers a result of Knoerrer from the commutative to the not necessarily commutative case. For this purpose we introduce the notion of ``dense subrings'' which seems rather technical but, nevertheless, useful. It enables the construction of ``almost versal'' families of modules for a given algebra and the definition of the ``number of parameters''. The semi--continuity implies, in particular, that the set of so-called ``wild algebras'' in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, together with the results of a former paper of the authors the semi-continuity implies that tame is indeed an open property for curve singularities (commutative CM rings). An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or finite dimensional bimodules.

Invariant syzygies and semisimple groups

The purpose of this paper is to explore some intriguing connections between Lie theory, relative homological algebra, invariant theory, and representations of finite dimensional algebras. In the process some results of Hilbert [20], Cartan and Eilenberg [7], Hochschild [21], Gelfand [ 171, and others are generalized, and a question of [Zl] is partly answered in Theorem 1.5. (Duflo seems to have stronger, unpublished, results on this problem.’ Sections 1 to 3 take place in a fairly general setting, whereas Section 4 focuses on semisimple Lie groups. The problem considered here goes back, in some sense, to Hilbert [20] (this paper was published 102 years ago!). To explain this, let me start by singling out two (out of many) fundamental contributions of the great mathematician. The first one is invariant theory, namely the study of the algebra (SV)n consisting of those polynomial functions on a finite dimensional vector space V* which are invariant under the action of a group H of linear transformations. The second one is the so-called syzygy theorem, which asserts, in today’s language, that the global dimension of the full symmetric algebra SV is dim V. This prompts the question: is there an algebra D, attached to V as naturally as SV is, on which H acts by automorphisms, such that the global dimension of the algebra B = DH of invariants is dim V? The most natural candidate is

Finite dimensional representations of algebras

Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.