Automorphism Groups Of Exterior Algebras Generated By Dynkin Graphs

The structure of a graph determines its automorphism group,and in turn,the automor-phism group of a graph can explain the properties of the graph.So determining the automor-phism group of a graph is an important subject in graph theory.In this thesis,we study the automorphism groups of exterior algebras generated by Dynkin graphs.According to L.Makar-Limanov’s result,if G is a finite gaph and K（G）is the exterior algebra generated by the graph,then Aut（K（G））is semi-direct product of LAut（K（G））and IAut（K（G））^[22],where LAut（K（G））is the subgroup generated by linear automorphisms that preserve the degrees,and IAut（K（G））is the normal subgroup generated by automorphisms such that g（v_i）=v_i+R_i,where v_iare vertices of G and R_i∈K（G）.We describe the linear automorphism subgroups LAut（K（G））of the exterior algebras gen-erated by Dynkin graphs at first.L.Makar-Limanov has proved that LAut（K（G））is the wreath product ofGL（v_i,K）and H^[22],where H is image ofφ:LAut（K（G））→Aut（G/～）,GL（v_i,K）is the general linear group of the equivalence class of v_iwhose order is the number of vertices equivalent to v_i.In this thesis,GL（v_i,K）and H are calculated respectively for the exterior algebras generated by Dynkin graphs of type A,D,E.Then the nonlinear automorphism subgroups IAut（K（G））of the exterior algebras gener-ated by Dynkin graphs of type A,D,E are studied.Since the group IAut（K（G））is generated by the central,triangular and inner automorphisms^[22].We described the central and inner auto-morphisms by calculating the central and invertible elements.We also calculate the triangular automorphisms of the exterior algebra generated by Dynkin graphs of A,D,E respectively.As a result,the nonlinear automorphism subgroups are expressed.