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.

Degrees in Auslander–Reiten components with almost split sequences of at most two middle termss

2015 ◽  
Vol 14 (07) ◽  
pp. 1550106 ◽  
Author(s):  
Claudia Chaio
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Finite Degree ◽  
Artin Algebra ◽  
Dimensional Algebra ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Almost Split Sequences ◽  
Irreducible Morphisms

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander–Reiten components Γ having only almost split sequences with at most two indecomposable middle terms, that is, α(Γ) ≤ 2. We prove that if f : X → Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z ∈ Γ and a morphism φ ∈ ℜn(Z, X)\ℜn+1(Z, X) for some positive integer n such that fφ = 0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f = (f1, f2)t : X → Y1 ⊕ Y2 is an irreducible epimorphism of finite left degree then we show that dl(f) = dl(f1) + dl(f2). We also characterize the artin algebras of finite representation type with α(ΓA) ≤ 2 in terms of a finite number of irreducible morphisms with finite degree.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

THE COINVARIANT ALGEBRA AND REPRESENTATION TYPES OF BLOCKS OF CATEGORY O

2001 ◽  
Vol 33 (6) ◽  
pp. 669-681 ◽  
Author(s):  
TH. BRÜSTLE ◽  
S. KÖNIG ◽  
V. MAZORCHUK
Keyword(s):  
Main Tool ◽  
Complex Numbers ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Semisimple Lie Algebra ◽  
Finite Dimensional ◽  
Category O ◽  
Representation Types ◽  
Coinvariant Algebra

Let [Gfr ] be a finite-dimensional semisimple Lie algebra over the complex numbers. Let A be the finite-dimensional algebra of a (regular or singular) block of the BGG-category [Oscr ] . By results of Soergel, A has a combinatorial description in terms of a subalgebra C0 of the coinvariant algebra C. König and Mazorchuk have constructed an embedding from C0-mod into the category [Fscr ](Δ) of A-modules having a Verma flag. This is the main tool for the classification of [Fscr ] (Δ) into finite, tame and wild representation types presented here. As a consequence a classification of A-mod into finite, tame and wild representation types is obtained, thus re-proving a recent result of Futorny, Nakano and Pollack.

scholarly journals EXT-FINITE MODULES FOR WEAKLY SYMMETRIC ALGEBRAS WITH RADICAL CUBE ZERO

2016 ◽  
Vol 102 (1) ◽  
pp. 108-121 ◽  
Author(s):  
KARIN ERDMANN
Keyword(s):  
Complete Classification ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Generation ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Weakly Symmetric ◽  
Support Varieties

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

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 Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

scholarly journals Duality and socle generators for residual intersections

2019 ◽  
Vol 2019 (756) ◽  
pp. 183-226 ◽  
Author(s):  
David Eisenbud ◽  
Bernd Ulrich
Keyword(s):  
Gorenstein Ring ◽  
Jacobian Determinant ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Duality Results ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Residue Theory

AbstractWe prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

Necessary conditions for power commuting in finite-dimensional algebras over a field

2018 ◽  
Vol 28 (5) ◽  
pp. 339-344
Author(s):  
Andrey V. Zyazin ◽  
Sergey Yu. Katyshev
Keyword(s):  
Necessary Conditions ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Finite Dimensional Algebras

Abstract Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.

scholarly journals Maximal subalgebras of finite-dimensional algebras

2019 ◽  
Vol 31 (5) ◽  
pp. 1283-1304 ◽  
Author(s):  
Miodrag Cristian Iovanov ◽  
Alexander Harris Sistko
Keyword(s):  
Associative Algebra ◽  
Complete Classification ◽  
Maximal Subalgebras ◽  
Incidence Algebras ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Closed Fields ◽  
Representations Of Algebras ◽  
Algebraically Closed Fields ◽  
Full Classification

AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

scholarly journals Residually finite dimensional algebras and polynomial almost identities

2020 ◽  
pp. 2250038
Author(s):  
Michael Larsen ◽  
Aner Shalev
Keyword(s):  
Lie Algebra ◽  
Finite Index ◽  
Open Group ◽  
Finite Codimension ◽  
Finite Dimensional Algebra ◽  
Nilpotent Ideal ◽  
Dimensional Algebra ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

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