Simply Connected, Adjoint and Universal Groups of Lie Type

The special linear group is the simply connected group and theprojective linear group is the adjoint group of Lie type A_n.They are distinguished sections of the (reductive) general lineargroup which certainly is of this type as well (root system).We shall characterize the general linear group as the universalgroup of type A_n. Indeed we shall introduce corresponding algebraicgroups and finite groups for each Lie type (to indecomposableroot systems). Knowledge of the universal group implies knowledgeof the related simply connected and adjoint groups; in certainrespects the universal group even appears to be better behaved(automorphisms, Schur multiplier, character table).