Rational Hilbert series of relatively free Jordan algebras

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Let K be a fixed field of characteristic zero. Assume V, is a k-dimensional vector space with a non-degenerate symmetric bilinear form over K and let Gk = V, + K be the Jordan algebra of this form [6]. Let V= V, be the vector space of countable dimension over K and set G = G,. Let 2Bk = var Gk and m = var G be the varieties of unitary Jordan algebras determined by the identities of the algebras Gk and G, respectively. The main result of this paper states that for every subvariety U of !IB the relatively free algebra FJ,(U) has a Hilbert (or Poincare) series which is a rational function. (This gives an affirmative answer to Problem No. 8.6 stated in [2].) As a consequence of the method of proof, an analogous result is established for the variety of pairs generated by the pair (C, V), where C denotes the Clifford algebra of I/.