Stanley, Laura (2011) Upper Triangular Matrices and Operations in Odd Primary Connective KTheory. PhD thesis, University of Sheffield.

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Abstract
Let $U_\infty\Z_p$ be the group of infinite invertible upper triangular matrices with entries in the $p$adic integers. Also let $\Aut_{\text{left}\ell\text{mod}}^0(\ell\wedge\ell)$ be the group of left $\ell$module automorphisms of $\ell\wedge\ell$ which induce the identity on mod $p$ homology, where $\ell$ is the Adams summand of the $p$adically complete connective $K$Theory spectrum. In this thesis we construct and prove there is an isomorphism between these two groups. We will then determine a specific matrix (up to conjugacy) which corresponds to the automorphism $1\wedge\psi^q$ of $\ell\wedge\ell$ where $\psi^q$ is the Adams operation and $q$ is an integer which generates the $p$adic units $\Z_p^\times$. We go on to look at the map $1\wedge\phi_n$ where $\phi_n=(\psi^q1)(\psi^qr)\cdots(\psi^qr^{n1})$ and $r=q^{p1}$ under a generalisation of the map which gave us the isomorphism. Lastly we use some of the ideas presented to give us a new way of looking at the ring of degree zero operations on the connective $p$local Adams summand via upper triangular matrices.
Item Type:  Thesis (PhD) 

Academic Units:  The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) 
Identification Number/EthosID:  uk.bl.ethos.555132 
Depositing User:  Miss Laura Stanley 
Date Deposited:  20 Jan 2012 15:14 
Last Modified:  27 Apr 2016 14:10 
URI:  http://etheses.whiterose.ac.uk/id/eprint/2015 