Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field

2008 ◽  
Vol 60 (4) ◽  
pp. 923-957 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto
Keyword(s):  
Height Function ◽  
Rational Functions ◽  
Trivial Extension ◽  
Closed Field ◽  
Quadratic Algebra ◽  
Endomorphism Algebra ◽  
Finite Dimensional ◽  
Free Rank ◽  
Rank 2 ◽  
Coordinate Rings

AbstractThe Kronecker modules , where m is a positive integer, h is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X over an algebraically closed field K, aremodels for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial f in K(X)[Y]. When the endomorphism algebra of is commutative and non-trivial, the regulator f must be quadratic in Y. If f has one repeated root in K(X), the endomorphismalgebra is the trivial extension for some vector space S. If f has distinct roots in K(X), then the endomorphisms forma structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End that are domains have zero radical. In addition, each semi-local End must be either a trivial extension or the product K × K.

Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

2007 ◽  
Vol 59 (1) ◽  
pp. 186-210 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto
Keyword(s):  
Quadratic Function ◽  
Height Function ◽  
Rational Functions ◽  
Closed Field ◽  
Height Functions ◽  
Planar Curve ◽  
Endomorphism Algebras ◽  
Form Part ◽  
Quadratic Function Field ◽  
Quadratic Function Fields

AbstractPurely simple Kronecker modules ℳ, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h: K ∪ ﹛∞﹜ → ﹛∞, 0, 1, 2, 3, … ﹜ is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X)[Y] called the regulator. When the module ℳ admits nontrivial endomorphisms, f must be linear or quadratic in Y. In that case ℳ is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End ℳ embeds in the quadratic function field K(X)[Y]/(f). For some height functions h of infinite support I, the search for a functional α for which (h, α) has regulator 0 comes down to having functions η : I → K such that no planar curve intersects the graph of η on a cofinite subset. If K has characterictic not 2, and the triplet (m, h, α) gives a purely-simple Kronecker module ℳ having non-trivial endomorphisms, then h attains the value ∞ at least once on K ∪ ﹛∞﹜ and h is finite-valued at least twice on K ∪ ﹛∞﹜. Conversely all these h form part of such triplets. The proof of this result hinges on the fact that a rational function r is a perfect square in K(X) if and only if r is a perfect square in the completions of K(X) with respect to all of its valuations.

COMPLEXITY OF TRIVIAL EXTENSIONS OF ITERATED TILTED ALGEBRAS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250067 ◽  
Author(s):  
MARJU PURIN
Keyword(s):  
Representation Type ◽  
Trivial Extension ◽  
Closed Field ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras

We study the complexity of a family of finite-dimensional self-injective k-algebras where k is an algebraically closed field. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H.

scholarly journals On the Hochschild coho-mology groups of endomorphism algebras of exceptional sequences

2004 ◽  
Vol 35 (3) ◽  
pp. 189-196
Author(s):  
Hailou Yao ◽  
Lihong Huang
Keyword(s):  
Associative Algebra ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Cohomology Groups ◽  
Endomorphism Algebra ◽  
Exceptional Sequence ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Exceptional Sequences

Let $ A $ be a finite dimensional associative algebra over an algebraically closed field $ k $, and $\mod A$ be the category of finite dimensional left $ A $-module and $ X_1,X_2,\ldots,X_n$ in $\mod A$ be a complete exceptional sequence, then we investigate the Hochschild Cohomology groups of endomorphism algebra of exceptional sequence $ {X_1,X_2, \ldots,X_n}$ in this paper.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

One-Dimensional Representations of the Cycle Subalgebra of a Semi-Simple Lie Algebra

1970 ◽  
Vol 13 (4) ◽  
pp. 463-467 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Cartan Subalgebra ◽  
Mass Function ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Noncommutative Algebra ◽  
One Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Mass Functions

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial.

scholarly journals Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

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.

Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 111-114
Author(s):  
F. Okoh
Keyword(s):  
Rational Functions ◽  
Indecomposable Module ◽  
Torsion Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
The Status ◽  
Hereditary Algebras ◽  
Divisible Modules ◽  
Torsion Free

AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

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