Geodesic
Growth in Poisson algebras
Algebra i logika, Tome 50 (2011) no. 1, pp. 68-88

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A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let $V$ be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities $$ \{\{x_1,y_1\},\{x_2,y_2\},\dots,\{x_m,y_m\}\}=0,\qquad\{x_1,y_1\}\cdot\{x_2,y_2\}\cdot\ldots\cdot\{x_m,y_m\}=0, $$ for some $m$. It is shown that the exponent of $V$ exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.
Mots-clés : Poisson algebra
Keywords: growth of variety, colength of variety. 
@article{AL_2011_50_1_a3,
     author = {S. M. Ratseev},
     title = {Growth in {Poisson} algebras},
     journal = {Algebra i logika},
     pages = {68--88},
     publisher = {mathdoc},
     volume = {50},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2011_50_1_a3/}
}
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S. M. Ratseev. Growth in Poisson algebras. Algebra i logika, Tome 50 (2011) no. 1, pp. 68-88. http://geodesic.mathdoc.fr/item/AL_2011_50_1_a3/