Associative and Jordan shift algebras

Abstract:

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 ${P_x}y = x{y^{\ast }}x$ where $^{\ast }$ is the involution on $R$ with ${u^{\ast }} = v$. As a consequence, we prove the existence of an infinite rectangular grid in a Jordan triple system $V$ containing tripotents $c$ and $d$ with ${V_2}(c) = {V_2}(d) \oplus ({V_2}(c) \cap {V_1}(d))$ and ${V_2}(c) \cap {V_1}(d) \ne 0$.