Codimension growth of special simple Jordan algebras

Giambruno, Antonio; Zaicev, Mikhail

doi:10.1090/S0002-9947-09-04865-X

Codimension growth of special simple Jordan algebras
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by Antonio Giambruno and Mikhail Zaicev PDF

Trans. Amer. Math. Soc. 362 (2010), 3107-3123 Request permission

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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