arXiv subjects

Fred Van Oystaeyen

Publications and source records attributed to Fred Van Oystaeyen.

22 records · Page 2Linked to original sources

arXiv2009

Dimension of Crystalline Graded Rings

The global dimension of a ring governs many useful abilities. For example, it is semi-simple if the global dimension is 0, hereditary if it is 1 and so on. We will calculate the global dimension of a Crystalline Graded Ring, as defined in the paper by E. Nauwelaerts and F. Van Oystaeyen, Introducing Crystalline Graded Algebras, Algebras and Representation Theory vol 11(2008), no. 2, 133--148.. We will apply this to derive a condition for the Crystalline Graded Ring to be semiprime. In the last section, we give a little bit of attention to the Krull-dimension.

math.RA↗

arXiv2009

Generalized Maximal Orders

Maximal Orders over an algebra are a generalization of the concept of a Dedekind domain. The definition given in Maximal Orders by Reiner, assumes that the field over which the algebra is defined is in the center of the order. Since we want to define maximal orders over a Crystalline Graded Ring (defined in Nauwelaerts, E.; Van Oystaeyen, F., Introducing crystalline graded algebras, Algebras and Representation Theory vol 11(2008), no. 2, 133--148.), this concept needs to be generalized. In this paper, we will weaken the condition that the field needs to be in the center, and still retain many of the desired properties of a maximal order.

math.SG↗

arXiv2009

Abelian Group Clifford Algebras

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to projective representations of a finite group (Abelian), $G$ say, such that the corresponding twisted group ring has minimal center. The latter then translates to the fact that the corresponding 2-cocycle allows a minimal (none!) number of ray classes and this forces a decomposition of $G$ in cyclic components in a suitable way, cf. Zmud, M., Symplectic geometries and projective representations of finite Abelian groups, (Russian) Mat. Sb. (N.S.) 87(129) (1972), 3--17.. In this small paper, I will provide a way to represent an Abelian Group Clifford Algebra using a matrix, and then give a way to calculate whether or not the center is trivial.

math.RA↗

arXiv2009

Maximal Graded Orders over Crystalline Graded Rings

Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. When the base ring is a commutative Dedekind domain, two constructions are given for producing maximal graded orders. On the way, a new concept appears, so-called, spectrally twisted group. Some general properties of it are studied. At the end of the paper several examples are considered.

math.RA↗