1 research outputs found
Module structure in certain algebras
For ages now, the literature has abounded with various graded algebras
whose homogeneous components can be treated as modules for the general
linear group (by algebra automorphisms) and the general linear Lie algebra
(by derivations). Most of these algebras are relatively free (for example,
polynomial algebras) but the exterior algebra of a vector space instances
one which is not. This paper is an attempt to treat these algebras in a
uniform manner, with particular emphasis on the module structure of their
components.
Aside from preliminaries, the thesis falls into three parts. The first
gives an abstract definition of the relevant algebras; this involves a mild
generalization of some concepts from Universal Algebra. The second
introduces the two actions above, but treats them independently of each
other. The final part brings the actions together by the process of
Chevalley reduction; here, the components are treated as modules for
certain distinguished subalgebras (first studied by J.E. Humphreys) of
Kostant's algebra.
It will be found that, roughly speaking, information regarding
composition structure is quite definitive (and algorithmically computable).
We also examine the problem of decomposing the components of particular
algebras (notably, the free Lie and Special Jordan algebras) in finite
characteristic