ASSOCIATIVE AND JORDAN SHIFT ALGEBRAS

Let R be the shift algebra, i.e., the associative algebra presented by generators u, v and the relation uv = 1. As N. Jacobson showed, R contains an infinite family of matrix units. In this paper, we describe the Jordan algebra R(+) and its unital special universal envelope by generators and relations. Moreover, we give a presentation for the Jordan triple system defined on R by P(x)y = xy*x where * is the involution on R with u* = v. As a consequence, we prove the existence of an infinite rectangular grid in a Jordan triple system V containing tripotents c and d with V-2(c) = V-2(d)+(V-2(C)boolean AND V-1(d)) and V-2(c)boolean AND V-1(d)not equal 0.