Sheila Brenner

Periodic algebras which are almost Koszul

The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such ‘almost Koszul’ algebras is developed and other examples are given.

Modular representations of p groups

Let G be an Abelian p group, k a field of characteristicp, and E a k algebra. Butler [3] has shown that, if G is elementary, noncyclic, and of order at least 16, then there exists a k(G) module whose endomorphism ring is a split extension of a nilpotent ideal by E. Hc uses this fact to prove the existence of modular representations of G with surprising pathologies. In this paper similar results are proved for Z, x Z, with p f 2, for Z, x 2, , and for Z, x 2, x 2, . The results therefore hold for any noncyclic p group with p + 2 (whether Abelian or not), and for any noncyclic Abelian 2 group of order eight or more. The finite dimensional modular representations of Zz x 2, are known [4] and do not fit into the same pattern. For example, let E’ be the truncated polynomial algebra over k’ (of characteristic 2) generated by commuting elements x and y which satisfy x 3 = y3 :m: 0. The endomorphism algebras of the finite dimensional indecomposable R’(Z, x 2,) modules may be Icalculated from Conlon’s list [4]. None is isomorphic to a split extension of a nilpotent ideal by E’. On the other hand, given an integer n, any finite dimensional k’ algebra E can be realized in this way with a h’(Z, x Z, X: Z,)[or Iz’(Z, x Z,)-] module of finite dimension greater than n. It seems probable that the non-Abelian 2 groups whose largest Abclian quotient is Za x Zz follow the pattern of Zz x Z, rather than that of the Abelian groups of order 8. Krugljak [8] has shown how, given inequivalent (finite dimensional) representations of any finite group G’ over a field K of characteristic p f2, it is possible to construct corresponding inequivalent representations of Z, x Z,, over h. Each such representation is indecomposable if and only if the corresponding representation of G’ is indecomposable. The constructions used in this paper may bc regarded as a simplification and extension of

Some modules with nearly prescribed endomorphism rings

It is now well known that many, even quite small, rings can have large indecomposable modules or modules which have pathological decomposition properties. (For references, see [3].) The technique for obtaining such results is to construct modules with endomorphism rings of prescribed type. In this paper the technique is applied to rings with lattices of ideals satisfying a simple condition. However, the results obtained do not depend on the lattice condition since similar results hold for the integers (and other rings) which do not satisfy the condition. The present paper generalises a result of Dickson and Kelly [9]. These authors show that a ring R in which the identity has a decomposition 1 = e, + e2 + ... + e, into orthogonal idempotents such that eiRei is a local ring (1 < i < n) and which has nondistributive lattice of ideals has, for each cardinal c < N, , indecomposable modules with minimal generating set of cardinal c. The work described here arose from a resemblance between certain structures occurring in the work of Dickson and Kelly and some which occur in the study of endomorphism algebras of a vector space with four or more distinguished subspaces [I]. The author is very grateful to Professors Dickson and Kelly for communicating their results prior to publication. In addition she thanks Professor E. R. Love and the University of Melbourne for their hospitality in the Mathematics Department of the University of Melbourne at the time this work was started.

On the kernel of an irreducible map

Let K be the kernel of an irreducible epimorphism f which is not a simk map. If the almost split sequence starting at K has decomposable middle term, then K is a simple module. This generalises a theorem of Krause which requires either the domain or range of f to be indecomposable.

Strong coupling calculation for the scattering of neutrons by carbon

Abstract The validity of the optical model and distorted wave approximation for the scattering of nucleons from a nucleus when the elastic and inelastic channels are strongly coupled is considered. The strong coupling method is applied to the scattering of 14 MeV neutrons from carbon and the results show that the coupling terms affect the shapes of both the elastic and inelastic differential cross sections. No systematic search for a best fit has been made as the complex potential used contains no spin-orbit term. A good fit to the data was obtained at forward angles, and the results suggest that, with a spin-orbit term included, the method could yield satisfactory results at all angles.

Irreducible maps and bilinear forms

Abstract Bautista showed in 1982 that the possible multiplicities of indecomposable summands of the domains and ranges of irreducible maps between modules over artin algebras are given by numerical invariants of certain bilinear forms associated with the algebra. We obtain further information about these multiplicities by relating the forms to those studied elsewhere in algebra and geometry. One spectacular result is that the allowable multiplicities for some algebras over the field of real numbers depend on J.F. Adams’ determination of the number of linearly independent vector fields on a sphere.