9 research outputs found
Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras
In this paper we show that each non-zero ideal of a twisted generalized Weyl
algebra (TGWA) intersects the centralizer of the distinguished subalgebra
in non-trivially. We also provide a necessary and sufficient condition
for the centralizer of in to be commutative, and give examples of TGWAs
associated to symmetric Cartan matrices satisfying this condition. By imposing
a certain finiteness condition on (weaker than Noetherianity) we are able
to make an Ore localization which turns out to be useful when investigating
simplicity of the TGWA. Under this mild assumption we obtain necessary and
sufficient conditions for the simplicity of TGWAs. We describe how this is
related to maximal commutativity of in and the (non-) existence of
non-trivial -invariant ideals of . Our result is a generalization of
the rank one case, obtained by D. A. Jordan in 1993. We illustrate our theorems
by considering some special classes of TGWAs and providing concrete examples.Comment: 32 pages, no figures, minor improvements of the presentation of the
materia
The Ideal Intersection Property for Groupoid Graded Rings
We show that if a groupoid graded ring has a certain nonzero ideal property,
then the commutant of the center of the principal component of the ring has the
ideal intersection property, that is it intersects nontrivially every nonzero
ideal of the ring. Furthermore, we show that for skew groupoid algebras with
commutative principal component, the principal component is maximal commutative
if and only if it has the ideal intersection property
Skew Category Algebras Associated with Partially Defined Dynamical Systems
We introduce partially defined dynamical systems defined on a topological
space. To each such system we associate a functor from a category to
\Top^{\op} and show that it defines what we call a skew category algebra . We study the connection between topological freeness of
and, on the one hand, ideal properties of and, on
the other hand, maximal commutativity of in . In
particular, we show that if is a groupoid and for each e \in \ob(G) the
group of all morphisms is countable and the topological space
is Tychonoff and Baire, then the following assertions are equivalent:
(i) is topologically free; (ii) has the ideal intersection property,
that is if is a nonzero ideal of , then ; (iii) the ring is a maximal abelian complex subalgebra of . Thereby, we generalize a result by Svensson, Silvestrov
and de Jeu from the additive group of integers to a large class of groupoids.Comment: 16 pages. This article is an improvement of, and hereby a replacement
for, version 1 (arXiv:1006.4776v1) entitled "Category Dynamical Systems and
Skew Category Algebras
Commutants and Centers in a 6-Parameter Family of Quadratically Linked Quantum Plane Algebras
We consider a family of associative algebras, defined as the quotient of a free algebra with the ideal generated by a set of multi-parameter deformed commutation relations between four generators consisting of five quantum plane relations between pairs of generators and one sub-quadratic relation inter-linking all four generators. For generic parameter vectors, the center and the commutants of the two of the generators are described and conditions on the parameters for these commutants to be itself commutative or non-commutative are obtained
Commutativity and ideals in category crossed products
In order to simultaneously generalize matrix rings and group graded crossed products, we introduce category crossed products. For such algebras we describe the centre and the commutant of the coefficient ring. We also investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptiness of intersections of the coefficient ring by nonzero two-sided ideals