68 research outputs found
Trivial Extensions of Gentle Algebras and Brauer Graph Algebras
We show that two well-studied classes of tame algebras coincide: namely, the
class of symmetric special biserial algebras coincides with the class of Brauer
graph algebras. We then explore the connection between gentle algebras and
symmetric special biserial algebras by explicitly determining the trivial
extension of a gentle algebra by its minimal injective co-generator. This is a
symmetric special biserial algebra and hence a Brauer graph algebra of which we
explicitly give the Brauer graph. We further show that a Brauer graph algebra
gives rise, via admissible cuts, to many gentle algebras and that the trivial
extension of a gentle algebra obtained via an admissible cut is the original
Brauer graph algebra.
As a consequence we prove that the trivial extension of a Jacobian algebra of
an ideal triangulation of a Riemann surface with marked points in the boundary
is isomorphic to the Brauer graph algebra with Brauer graph given by the arcs
of the triangulation.Comment: Minor changes, to appear in Journal of Algebr
On the Hochschild cohomology of tame Hecke algebras
We explicitly calculate a projective bimodule resolution for a special
biserial algebra giving rise to the Hecke algebra H_q(S_4) when q=-1. We then
determine the dimensions of the Hochschild cohomology groups.Comment: Changes made to introduction and final sectio
Chebyshev polynomials on symmetric matrices
In this paper we evaluate Chebyshev polynomials of the second-kind on a class
of symmetric integer matrices, namely on adjacency matrices of simply laced
Dynkin and extended Dynkin diagrams. As an application of these results we
explicitly calculate minimal projective resolutions of simple modules of
symmetric algebras with radical cube zero that are of finite and tame
representation type
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
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
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
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