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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Associative and Jordan shift algebras

Authors: Ottmar Loos and Erhard Neher
Journal: Proc. Amer. Math. Soc. 120 (1994), 27-36
MSC: Primary 17C65; Secondary 16S99
MathSciNet review: 1158003
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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$.

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Article copyright: © Copyright 1994 American Mathematical Society