Let R be a commutative domain of stable range 1 with 2 a unit. In this paper we describe the homomorphisms between SL 2 (R) and GL 2 (K) where K is an algebraically closed field. We show that every non-trivial homomorphism can be decomposed uniquely as a product of an inner automorphism and a homomorphism induced by a morphism between R and K. We also describe the homomorphisms between GL 2 (R) and GL 2 (K). Those homomorphisms are found of either extensions of homomorphisms from SL 2 (R) to GL 2 (K) or the products of inner automorphisms with certain group homomorphisms from GL 2 (R) to K.
Abstract. This paper is about some graph isomorphisms between the Auslander-Reiten quivers of the path algebras of quivers with underlying Dynkin diagrams of type A l , and the weight diagrams or the weight graphs relative to some basic (adjoint) representations of a semi-simple complex Lie algebra associated to the same Dynkin diagrams. The crucial point is that, in the attempt to establish the above relationships, we constructed the oriented weight diagrams and the oriented weight graphs with respect to particular orientations on the Dynkin diagrams of type A l . Our main goal is to furnish another application of weight theory and visual representations.
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