scholarly journals Subring Depth, Frobenius Extensions, and Towers

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Lars Kadison
Keyword(s):  
Modular Function ◽  
Finite Representation ◽  
Finite Dimensional ◽  
Frobenius Extension ◽  
Ring Structures ◽  
Bar Resolution ◽  
Minimum Depth ◽  
Left And Right ◽  
Special Ring ◽  
Tensor Powers

The minimum depthd(B,A)of a subringB⊆Aintroduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show thatd(B,A)< ∞ ifAis a finite-dimensional algebra andBehas finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. IfA⊇Bis a QF extension, minimum left and right even subring depths are shown to coincide. IfA⊇Bis a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depthnextensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.

Hereditary artinian rings of finite representation type and extensions of simple artinian rings

1987 ◽  
Vol 102 (3) ◽  
pp. 411-420 ◽  
Author(s):  
Aidan Schofield
Keyword(s):  
Artinian Ring ◽  
Representation Type ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Finite Dimensional ◽  
Coxeter Diagram ◽  
Skew Fields ◽  
Left And Right ◽  
Natural Way

In [1], Dowbor, Ringel and Simson consider hereditary artinian rings of finite representation type; it is known that if A is an hereditary artinian algebra of finite representation type, finite-dimensional over a field, then it corresponds to a Dynkin diagram in a natural way; they show that an hereditary artinian ring of finite representation type corresponds to a Coxeter diagram. However, in order to construct an hereditary artinian ring of finite representation type corresponding to a Coxeter diagram that is not Dynkin, they show that it is necessary though not sufficient to find an extension of skew fields such that the left and right dimensions are both finite but are different. No examples of such skew fields were known at the time. In [3], I constructed such extensions, and the main aim of this paper is to extend the methods of that paper to construct an extension of skew fields having all the properties needed to construct an hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5).

scholarly journals Algebras of bounded finite dimensional representation type

1995 ◽  
Vol 37 (3) ◽  
pp. 289-302 ◽  
Author(s):  
Allen D. Bell ◽  
K. R. Goodearl
Keyword(s):  
Representation Type ◽  
Dimensional Representation ◽  
Dimensional Algebra ◽  
Finite Dimensional Representation ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Left And Right ◽  
Bounded Representation ◽  
Algebra Over A Field

It is well known that for finite dimensional algebras, “bounded representation type” implies “finite representation type”; this is the assertion of the First Brauer-Thrall Conjecture (hereafter referred to as Brauer-Thrall I), proved by Roiter [26] (see also [23]). More precisely, it states that if R is a finite dimensional algebra over a field k, such that there is a finite upper bound on the k-dimensions of the finite dimensional indecomposable right R-modules, then up to isomorphism R has only finitely many (finite dimensional) indecomposable right modules. The hypothesis and conclusion are of course left-right symmetric in this situation, because of the duality between finite dimensional left and right R-modules, given by Homk(−, k). Furthermore, it follows from finite representation type that all indecomposable R modules are finite dimensional [25].

scholarly journals Fields of Definition for Representations of Associative Algebras

2018 ◽  
Vol 62 (1) ◽  
pp. 291-304
Author(s):  
Dave Benson ◽  
Zinovy Reichstein
Keyword(s):  
Finite Extension ◽  
Representation Type ◽  
Essential Dimension ◽  
Extension Field ◽  
Associative Algebras ◽  
Separable Extension ◽  
Finite Representation ◽  
Field Of Definition ◽  
Finite Dimensional ◽  
A Finite Group

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Additive maps having a generalized derivation expansion

2015 ◽  
Vol 14 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Additive Map ◽  
Finite Dimensional ◽  
Left And Right ◽  
Central Simple ◽  
Skolem Theorem ◽  
Additive Maps

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

scholarly journals Complete slices and homological properties of tilted algebras

1994 ◽  
Vol 36 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Ibrahim Assem ◽  
Flávio Ulhoa Coelho
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Homological Dimensions ◽  
The Subject ◽  
Left And Right ◽  
Almost All

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

Artin's problem for skew field extensions

1985 ◽  
Vol 97 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. H. Schofield
Keyword(s):  
Artinian Ring ◽  
Field Extension ◽  
Subsequent Paper ◽  
Commutative Field ◽  
Finite Representation ◽  
New Approach ◽  
Skew Field ◽  
Skew Fields ◽  
Left And Right ◽  
The Right

For a commutative field extension, L ⊃ K, it is clear that a left basis of L over K; is also a right basis of L over K; however, for an extension of skew fields, this may easily fail, though it is hard to determine whether the right and left dimension may be different. Cohn ([4], ch. 5), however, was able to find extensions of skew fields such that the left and right dimensions were an arbitrary pair of cardinals subject only to the restrictions that neither were 1 and at least one of them was infinite. In this paper, I shall present a new approach that allows us to construct extensions of skew fields such that the left and right dimensions are arbitrary integers not equal to 1. In a subsequent paper, [7], I shall present related results and consequences; in particular, there is a construction of a hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5) answering the question raised by Dowbor, Ringel and Simson[5].

WEYL ALGEBRAS, FOURIER TRANSFORMATIONS AND INTEGRALS ON FINITE-DIMENSIONAL HOPF ALGEBRAS

1994 ◽  
Vol 06 (01) ◽  
pp. 149-166 ◽  
Author(s):  
FLORIAN NILL
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Multiplication Operators ◽  
Hermitian Forms ◽  
Weyl Algebras ◽  
Finite Dimensional ◽  
Natural Choice ◽  
Translation Operators ◽  
Canonical Representations ◽  
Left And Right

The "Weyl algebras" [Formula: see text], σ, σ′ ∈ {+, –}, over a finite-dimensional Hopf algebra H are defined to be the abstract algebras generated by left or right (σ = ±) multiplication operators Qσ (ψ), ψ ∈ H, and left or right (σ′ = ±) translation operators Pσ′ (a), [Formula: see text] dual of H. It is shown that these algebras are all isomorphic to End (H). Fourier transformations are defined as intertwiners [Formula: see text] implementing a natural isomorphism [Formula: see text]. As a byproduct, this formalism provides a new proof of the invertibility of the antipode and the uniqueness (up to multiplication by a constant) of left and right integrals on finite-dimensional Hopf algebras. If H is a star Hopf algebra, the canonical representations of [Formula: see text] become star representation and Fourier transformation becomes an isometry w.r.t. a natural choice of nondegenerate hermitian forms on H and [Formula: see text]. Some comments on the relation with the C*-approach of Woronowicz and Podleś & Woronowicz are added.

scholarly journals Representation-Directed Diamonds

2001 ◽  
Vol 4 ◽  
pp. 14-21
Author(s):  
Peter Dräxler
Keyword(s):  
Finite Dimensional Algebra ◽  
Program System ◽  
Dimensional Algebra ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Positive Roots ◽  
Algebraically Closed Fields

AbstractA module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

scholarly journals PURITY RELATIVE TO CLASSES OF FINITELY PRESENTED MODULES

2013 ◽  
Vol 12 (08) ◽  
pp. 1350050 ◽  
Author(s):  
AKEEL RAMADAN MEHDI
Keyword(s):  
General Description ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Left And Right ◽  
Finitely Presented ◽  
Tame Hereditary Algebras ◽  
Finite Dimensional Algebras

We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure-injectives which we make completely explicit in the case of tame hereditary algebras.

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