Polynomial Identities of Algebras of Small Dimension |
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| Authors: | Antonio Giambruno Sergey Mishchenko Mikhail Zaicev |
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| Affiliation: | 1. Department of Mathematics and Applications , University of Palermo , Palermo , Italy agiambr@unipa.it;3. Department of Algebra and Geometric Computations, Faculty of Mathematics and Mechanics , Ulyanovsk State University , Ulyanovsk , Russia;4. Department of Algebra, Faculty of Mathematics and Mechanics , Moscow State University , Moscow , Russia |
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| Abstract: | It is well known that given an associative algebra or a Lie algebra A, its codimension sequence c n (A) is either polynomially bounded or grows at least as fast as 2 n . In [2 Giambruno , A. , Mishchenko , S. , Zaicev , M. ( 2006 ). Algebras with intermediate growth of the codimensions . Adv. Appl. Math. 37 ( 3 ): 360 – 377 .[Crossref], [Web of Science ®] , [Google Scholar]] we proved that for a finite dimensional (in general nonassociative) algebra A, dim A = d, the sequence c n (A) is also polynomially bounded or c n (A) ≥ a n asymptotically, for some real number a > 1 which might be less than 2. Nevertheless, for d = 2, we may take a = 2. Here we prove that for d = 3 the same conclusion holds. We also construct a five-dimensional algebra A with c n (A) < 2 n . |
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| Keywords: | Codimensions Exponential growth Polynomial identity |
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