5,488 research outputs found
Support Varieties and Representation Type of Self-Injective Algebras
We use the theory of varieties for modules arising from Hochschild cohomology
to give an alternative version of the wildness criterion of Bergh and Solberg:
If a finite dimensional self-injective algebra has a module of complexity at
least 3 and satisfies some finiteness assumptions on Hochschild cohomology,
then the algebra is wild. We show directly how this is related to the analogous
theory for Hopf algebras that we developed. We give applications to many
different types of algebras: Hecke algebras, reduced universal enveloping
algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.Comment: 21 page
Some Problems in the Representation Theory of Simple Modular Lie Algebras
The finite-dimensional restricted simple Lie algebras of characteristic p > 5
are classical or of Cartan type. The classical algebras are analogues of the
simple complex Lie algebras and have a well-advanced representation theory with
important connections to Kazhdan-Lusztig theory, quantum groups at roots of
unity, and the representation theory of algebraic groups. We survey progress
that has been made towards developing a representation theory for the
restricted simple Cartan-type Lie algebras, discuss comparable results in the
classical case, formulate a couple of conjectures, and pose a dozen open
problems for further study.Comment: References updated; a few minor changes made in this versio
Representation theory and projective geometry
We give an elementary introduction to our papers relating the geometry of
rational homogeneous varieties to representation theory. We also describe
related work and recent progress.Comment: 37 pages with picture
Support varieties and representation type of small quantum groups
In this paper we provide a wildness criterion for any finite dimensional Hopf
algebra with finitely generated cohomology. This generalizes a result of
Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields
of arbitrary characteristic. Our proof uses the theory of support varieties for
modules, one of the crucial ingredients being a tensor product property for
some special modules. As an application we prove a conjecture of Cibils stating
that small quantum groups of rank at least two are wild.Comment: 14 pages; minor revisions; to appear in Int. Math. Res. No
Action Type Geometrical Equivalence of Representations of Groups
For every variety of algebras and every algebras in these variety we can
consider an algebraic geometry. Algebras may be many sorted (not necessarily
one sorted) algebras. A set of sorts is fixed for each variety. This theory can
be applied to the variety of representations of groups over fixed commutative
ring with unit. We consider a representation as two sorted algebra. We
concentrate on the case of the action type algebraic geometry of
representations of groups. In this case algebraic sets are defined by systems
of action type equations and equations in the acting group are not considered.
This is the special case, which cannot be deduced from the general theory. In
this paper the following basic notions are introduced: action type geometrical
equivalence of two representations, action type quasi-identity in
representations, action type quasi-variety of representations, action type
Noetherian variety of representations, action type geometrically Noetherian
representation, action type logically Noetherian representation.Comment: 35 page
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