185 research outputs found
Universal deformation rings of modules over Frobenius algebras
Let be a field, and let be a finite dimensional -algebra. We
prove that if is a self-injective algebra, then every finitely
generated -module whose stable endomorphism ring is isomorphic to
has a universal deformation ring which is a complete local
commutative Noetherian -algebra with residue field . If is also
a Frobenius algebra, we show that is stable under taking
syzygies. We investigate a particular Frobenius algebra of dihedral
type, as introduced by Erdmann, and we determine for every
finitely generated -module whose stable endomorphism ring is
isomorphic to .Comment: 25 pages, 2 figures. Some typos have been fixed, the outline of the
paper has been changed to improve readabilit
Invariant forms on irreducible modules of simple algebraic groups
Let be a simple linear algebraic group over an algebraically closed field
of characteristic and let be an irreducible rational
-module with highest weight . When is self-dual, a basic
question to ask is whether has a non-degenerate -invariant alternating
bilinear form or a non-degenerate -invariant quadratic form.
If , the answer is well known and easily described in terms of
. In the case where , we know that if is self-dual, it
always has a non-degenerate -invariant alternating bilinear form. However,
determining when has a non-degenerate -invariant quadratic form is a
classical problem that still remains open. We solve the problem in the case
where is of classical type and is a fundamental highest weight
, and in the case where is of type and for . We also give a solution in some specific
cases when is of exceptional type.
As an application of our results, we refine Seitz's description of
maximal subgroups of simple algebraic groups of classical type. One consequence
of this is the following result. If are simple
algebraic groups and is irreducible, then one of the following
holds: (1) is not self-dual; (2) both or neither of the
modules and have a non-degenerate invariant
quadratic form; (3) , , and .Comment: 46 pages; to appear in J. Algebr
Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth
Let be an arbitrary field, be a -algebra, and be a
-module. When it exists, the universal deformation ring
of is a -algebra whose local homomorphisms from to
parametrize the lifts of up to , where is any appropriate
complete, local commutative Noetherian -algebra. Symmetric special biserial
algebras, which coincide with Brauer graph algebras, can be viewed as
generalizing the blocks of finite type -modular group algebras. Bleher and
Wackwitz classified the universal deformation rings for all modules for
symmetric special biserial algebras with finite representation type. In this
paper, we begin to address the tame case. Specifically, let be a
symmetric special biserial algebra of polynomial growth which coincides with an
acyclic Brauer graph algebra. We classify the universal deformation rings for
those -modules with stable endomorphism ring isomorphic to .
The latter is a natural condition, since it guarantees the existence of the
universal deformation ring
On -filtrations of Weyl modules
This paper considers Weyl modules for a simple, simply connected algebraic
group over an algebraically closed field of positive characteristic
. The main result proves, if (where is the Coxeter
number) and if the Lusztig character formula holds for all (irreducible modules
with) regular restricted highest weights, then any Weyl module
has a -filtration, namely, a filtration with
sections of the form
, where is
restricted and is arbitrary dominant. In case the highest weight
of the Weyl module is -regular, the
-filtration is compatible with the -radical series of the module. The
problem of showing that Weyl modules have -filtrations was first
proposed as a worthwhile ("w\"unschenswert") problem in Jantzen's 1980 Crelle
paper.Comment: Latest version corrects minor mistakes in previous versions. A
reference is made to Williamson's recent arXiv posting, providing some
relevant discussion in a footnote. [Comments on earlier versions: Previous v.
1 with minor errors and statements corrected. Improved organization. Should
replace v. 2 which is an older version (even older than v.1) and was
mistakenly posted.
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