3,322 research outputs found
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.Comment: Minor revision, to appear in Advances in Mathematic
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.Comment: Minor corrections, to appear in Bulletin des Sciences Mathematique
Group actions and coverings of Brauer graph algebras
We develop a theory of group actions and coverings on Brauer graphs that
parallels the theory of group actions and coverings of algebras. In particular,
we show that any Brauer graph can be covered by a tower of coverings of Brauer
graphs such that the topmost covering has multiplicity function identically
one, no loops, and no multiple edges. Furthermore, we classify the coverings of
Brauer graph algebras that are again Brauer graph algebras.Comment: 26 pages Correction to statement of Theorem 6.7; a tower of coverings
has been introduce
On Artin algebras arising from Morita contexts
We study Morita rings \Lambda_{(\phi,\psi)}=\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 and are zero. Further we give
bounds for the global dimension of a Morita ring , regarded as
an Artin algebra, in terms of the global dimensions of and in the case
when both and 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 , where is an Artin algebra.Comment: 29 pages, revised versio
From Monomials to Words to graphs
Given a finite alphabet X and an ordering on the letters, the map \sigma
sends 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 generated by \sigma(I) in the free monoid
is finitely generated. Whether there exists an ordering such that
is finitely generated turns out to be NP-complete. The latter problem is
closely related to the recognition problem for comparability graphs.Comment: 27 pages, 2 postscript figures, uses gastex.st
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