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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Codimension growth of special simple Jordan algebras


Authors: Antonio Giambruno and Mikhail Zaicev
Journal: Trans. Amer. Math. Soc. 362 (2010), 3107-3123
MSC (2000): Primary 17C05, 16P90; Secondary 16R10
Published electronically: December 22, 2009
MathSciNet review: 2592948
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Abstract: Let $ R$ be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial $ f$ multialternating on disjoint sets of variables which is not a polynomial identity of $ R$. We then study the growth of the polynomial identities of the Jordan algebra $ R$ through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial $ f$, we are able to compute the exponential rate of growth of the sequence of Jordan codimensions of $ R$ and prove that it equals the dimension of the Jordan algebra over its center. We also show that for any finite dimensional special Jordan algebra, such an exponential rate of growth cannot be strictly between $ 1$ and $ 2$.


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Additional Information

Antonio Giambruno
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Email: a.giambruno@unipa.it

Mikhail Zaicev
Affiliation: Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992 Russia
Email: zaicev@mech.math.msu.su

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04865-X
PII: S 0002-9947(09)04865-X
Keywords: Polynomial identity, special Jordan algebra, codimensions, exponential growth
Received by editor(s): April 16, 2008
Published electronically: December 22, 2009
Additional Notes: The first author was partially supported by MIUR of Italy
The second author was partially supported by RFBR grant No. 06-01-00485 and SSC-5666.2006.1
Article copyright: © Copyright 2009 American Mathematical Society