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Automorphism groups of minimal algebras

University of British Columbia

Rational homotopy theory is the study of uniquely divisible homotopy invariants. For each nilpotent space X the association X ——» minimal algebra for X is a complete determination of these invariants. If X is a space and Mx its minimal algebra, the algebraic group Aut Mx and the representation Aut Mx ——» Gl(Mx) have considerable influence on the structure of Mx . This thesis contains a systematic study of this interaction. Chapter I contains preliminary results from algebraic group theory and general topology. In Chapter II I define and study inverse limits of algebraic groups. I prove that many of the known structural properties of algebraic groups remain valid in this more general setting. Emphasis is placed on the conjugacy theorems that are particularly useful for studying minimal algebras. Chapter III is the main part of the thesis where I develop a structure theory for minimal algebras which relates toroidal symmetry to retracts. Precisely, if M is a minimal algebra, then there exists a 1-parameter subgroup λ : Q* ——> Aut Mx such that λ extends to λ : Q——» End Mx λ: (0) = e = e²: Mx——» Mx Further if e so chosen is minimal then it is uniquely determined up to conjugation by Aut Mx . In the interesting case where e = 0m I give a pro-algebraic group theoretic proof of uniqueness of coproduct and product decompositions in the appropriate homotopy category.

Text

2010-02-25

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http://hdl.handle.net/2429/20897

Mathematics

University of British Columbia

Graduate