M.Sc Thesis
M.Sc Student Adi Wolf Automorphisms and Forms of Path Algebras Department of Mathematics Full Professor Aljadeff Eli

Abstract

Let K/k be a field extension and A a finite dimensional k-algebra. We consider the question of classifying the K/k-forms of A, i.e., find up to k-isomorphism all the k-algebras B such that AkKBkK. When K/k is a Galois extension and A=Mn(k) or A=kn these are known to be the central simple k-algebras of degree n and the étale k-algebras of dimension n, respectively. We would like to consider some algebras with non-trivial Jacobson radicals, namely, hereditary path algebras on strongly acyclic quivers. An algebra A is called elementary if A/J kn, where J is the Jacobson radical of A. Gabriel's theorem (1972) shows that when k is a perfect field, every hereditary elementary algebra is isomorphic to a path algebra on some acyclic quiver. Recall that when K/k is a Galois extension, the so called descent theory enables us to interpret each K/k-form of A as an algebra of invariants under some action of Gal(K/k) on AkK. These actions are classified by the pointed set of 1-cocycles H1(Gal(K/k), AutK-alg(AkK)). Assume A is a hereditary path algebra on a strongly acyclic quiver Γ. Then every automorphism of the quiver induces an automorphism of the path algebra. We show that when char(k)=0 and K/k is a finite Galois extension, the K/k-forms of A are classified by the cohomology pointed set H1(Gal(K/k), SΓ) where SΓ is a certain subgroup of automorphisms of the quiver. When Γ is a rooted tree, this translates the classification of K/k-forms of into a combinatorial problem. We define the notion of forms of a rooted tree Γ and of evaluations, which are certain algebras these combinatorial forms give rise to, and show that the K/k-forms of are classified by evaluations of forms of Γ.