by Bovdi, Victor; Juhász, Tibor; Spinelli, Ernesto

Let $K$ be a field of positive characteristic $p$ and $KG$ the group algebra of a group $G$. It is known that, if $KG$ is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most $|G^{\, \prime}|+1$, where $|G^{\, \prime}|$ is the order of the commutator subgroup. The authors previously determined those groups $G$ for which this index is maximal and here they determine the groups $G$ for which it is `almost maximal', that is, it takes the next highest possible value, namely $|G^{\, \prime}|-p+2$.

DOI: 10.1007/s10468-006-9022-5
Online Date: 5/25/2006
Print publication date: 6/1/2006
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by Rump, Wolfgang

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal.

DOI: 10.1007/s10468-006-9006-5
Online Date: 5/20/2006
Print publication date: 6/1/2006
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by Brandl, M. Katherine

We examine families of twists by an automorphism of the complex polynomial ring on n generators. The multiplication in the twisted algebra determines a Poisson structure on affine n-space. We demonstrate that if the automorphism has a single eigenvalue, then the primitive ideals in the twist are parameterized by the algebraic symplectic leaves associated to this Poisson structure. Furthermore, in this case all of the leaves are algebraic and can be realized as the orbits of an algebraic group.

DOI: 10.1007/s10468-006-9008-3
Online Date: 5/3/2006
Print publication date: 6/1/2006
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