Partially commutative and differential graded algebraic structures

Abstract

The objects of study in this thesis are partially commutative and di erential graded algebraic structures. In fact my thesis is in two parts. The rst on partially commutative algebraic structures is concerned with automorphism groups of par- tially commutative groups and their nite presentations. The second on di erential graded algebraic structures is concerned with di erential graded modules. I have given a description for Aut(G􀀀 ), the automorphism group of the partially commutative group G􀀀 following Day's work, where 􀀀 is a nite simple graph. I have given a description for the subgroup Conj(G􀀀 ) of automorphism group Aut(G􀀀 ) following Toinet's work. We have found a nite presentation for the subgroup ConjV of the automorphism group Aut(G􀀀 ). I have developed AutParCommGrp (Finite Presentations of Automor- phism Groups of Partially Commutative Groups and Their Subgroups) a package using the GAP system for computation of a nite presentation for Aut(G􀀀 ), Conj(G􀀀 ) and ConjV respectively. In the second part of the thesis we consider the following situation: Let K be a eld of characteristic two and let R = K[x1; x2; ; xn] be a graded polynomial ring, graded in the negative way. Suppose M is a di erential graded R-module with di erential @ of degree P. We have constructed a classi cation for some types of di erential graded R-module where P 􀀀2, n > 1. This classi cation gives a partial algorithm to test whether such modules are solvable. For modules outside the classi cation we cannot decide, using our methods, whether or not they are solvable. Also, we have proved in one case that M is solvable when R is a graded polynomial ring, graded in the usual way (non-negatively graded) with (P 2, n > 1). We have developed an algorithm and written a GAP package SDGM (Solvable Di erential Graded Modules) to check whether the di erential graded R-module M with di erential @ of degree P is solvable or not. Documentation has been written for all the packages above.