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Edward L. Green

Publications and source records attributed to Edward L. Green.

18 recordsLinked to original sources
arXiv2023

The Commuting Algebra

Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is a field. We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generated by all differences of parallel paths in $Q$. We show that $KQ/C$ is always finite dimensional and its global dimension is finite. Furthermore, we prove that $KQ/C$ is Morita equivalent to an incidence algebra. The paper starts with the more general setting, where $KQ$ is replaced by $KQ/I$ with $I$ a two-sided ideal in $KQ$.

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arXiv2019

Algebras and varieties

In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The case of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.

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arXiv2019

On quasi-hereditary algebras

Establishing whether an algebra is quasi-hereditary or not is, in general, a difficult problem. In this paper we introduce a sufficient criterion to determine whether a general finite dimensional algebra is quasi-hereditary by showing that the question can be reduced to showing that a closely associated monomial algebra is quasi-hereditary. For monomial algebras, we give an explicit, easily verifiable, necessary and sufficient criterion to determine whether it is quasi-hereditary.

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arXiv2018

Reduction techniques for the finitistic dimension

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.

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arXiv2017

Almost gentle algebras and their trivial extensions

In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one, is the trivial extension of an almost gentle algebra. We show that to every almost gentle algebra A is associated a hypergraph, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Amongst other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

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arXiv2017

Brauer configuration algebras: A generalization of Brauer graph algebras

In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying and analysing structural properties of Brauer configurations and Brauer configuration algebras, we show that a Brauer configuration algebra is multiserial; that is, its Jacobson radical is a sum of uniserial modules whose pairwise intersection is either zero or a simple module. The paper ends with a detailed study of the relationship between radical cubed zero Brauer configuration algebras, symmetric matrices with non-negative integer entries, finite graphs and associated symmetric radical cubed zero algebras.

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arXiv2017

Convex subquivers and the finitistic dimension

Let $\cQ$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\cQ$ and quotients of the path algebra $K\cQ$. We introduce the homological heart of $\cQ$ which is a particularly nice convex subquiver of $\cQ$. For any algebra of the form $K\cQ/I$, the algebra associated to $K\cQ/I$ and the homological heart have similar homological properties. We give an application showing that the finitistic dimension conjecture need only be proved for algebras with path connected quivers.

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arXiv2017

Special multiserial algebras, Brauer configuration algebras and more : a survey

We survey results on multiserial algebras, special multiserial algebras and Brauer configuration algebras. A structural property of modules over a special multiserial algebra is presented. Almost gentle algebras are introduced and we describe some results related to this class of algebras. We also report on the structure of radical cubed zero symmetric algebras.

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arXiv2017

The Geometry of Strong Koszul Algebras

Koszul algebras with quadratic Groebner bases, called strong Koszul algebras, are studied. We introduce affine algebraic varieties whose points are in one-to-one correspondence with certain strong Koszul algebras and we investigate the connection between the varieties and the algebras.

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arXiv2016

Multiserial and special multiserial algebras and their representations

In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module. We show that all finitely generated modules over a special multiserial algebra are multiserial. In particular, this implies that, in analogy to special biserial algebras being biserial, special multiserial algebras are multiserial. We then show that the class of symmetric special multiserial algebras coincides with the class of Brauer configuration algebras, where the latter are a generalization of Brauer graph algebras. We end by showing that any symmetric algebra with radical cube zero is special multiserial and so, in particular, it is a Brauer configuration algebra.

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arXiv2014

On the diagonal subalgebra of an Ext algebra

Let $R$ be a Koszul algebra over a field $k$ and $M$ be a linear $R$-module. We study a graded subalgebra $Δ_M$ of the Ext-algebra $\operatorname{Ext}_R^*(M,M)$ called the diagonal subalgebra and its properties. Applications to the Hochschild cohomology ring of $R$ and to periodicity of linear modules are given. Viewing $R$ as a linear module over its enveloping algebra, we also show that $Δ_R$ is isomorphic to the graded center of the Koszul dual of $R$.

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arXiv2013

On Artin algebras arising from Morita contexts

We study Morita rings $Λ_{(ϕ,ψ)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$ in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $ϕ$ and $ψ$ are zero. Further we give bounds for the global dimension of a Morita ring $Λ_{(0,0)}$, regarded as an Artin algebra, in terms of the global dimensions of $A$ and $B$ in the case when both $ϕ$ and $ψ$ are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring with $A=N=M=B=Λ$, where $Λ$ is an Artin algebra.

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arXiv2008

$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of degree $d$ is $d$-Koszul if and only if its associated monomial algebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class 2-$d$-Koszul algebras are introduced. In particular, it shown that 2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.

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arXiv2005

An algorithmic approach to resolutions

We provide an algorithmic method for constructing projective resolutions of modules over quotients of path algebras. This algorithm is modified to construct minimal projective resolutions of linear modules over Koszul algebras.

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arXiv2005

Multiplicative structures for Koszul algebras

Let $Λ=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $Λ$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $Λ$ over the enveloping algebra $Λ^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $Ł$ relative to the resolution $\mathbb{P}$ is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of $Ł$ with respect to a canonical basis over $k$ associated to the resolution $\mathbb{F}$. The natural map from the Hochschild cohomology to the Koszul dual of $Λ$ is shown to be surjective onto the graded centre of the Koszul dual.

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arXiv2004

Unified Field Theory From Enlarged Transformation Group. The Consistent Hamiltonian

A theory has been presented previously in which the geometrical structure of a real four-dimensional space time manifold is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group. The group enlargement was accomplished by including those transformations to anholonomic coordinates under which conservation laws are covariant statements. Field equations have been obtained from a variational principle which is invariant under the larger group. These field equations imply the validity of the Einstein equations of general relativity with a stress-energy tensor that is just what one expects for the electroweak field and associated currents. In this paper, as a first step toward quantization, a consistent Hamiltonian for the theory is obtained. Some concluding remarks are given concerning the need for further development of the theory. These remarks include discussion of a possible method for extending the theory to include the strong interaction.

gr-qc↗
arXiv2003

From Monomials to Words to graphs

Given a finite alphabet X and an ordering on the letters, the map σsends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal <σ(I)> generated by σ(I) in the free monoid is finitely generated. Whether there exists an ordering such that <σ(I)> is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.

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