Abstract
We consider associative algebras presented by a finite set of generators and relations of special form: each generator is annihilated by some polynomial, and the sum of generators is zero. The growth of this algebra in dependence on the degrees of the polynomials annihilating the generators is studied. The tuples of degrees for which the algebras are finite-dimensional, have polynomial growth, or have exponential growth are indicated. To the tuple of degrees, we assign a graph, and the above-mentioned cases correspond to Dynkin diagrams, extended Dynkin diagrams, and the other graphs, respectively. For extended Dynkin diagrams, we indicate the hyperplane in the space of parameters (roots of the polynomials) on which the corresponding algebras satisfy polynomial identities.
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Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 39, No. 3, pp. 14–27, 2005
Original Russian Text Copyright © by A. S. Mellit, Yu. S. Samoilenko, and M. A. Vlasenko
Partially supported by DFG grant no. 436 UKR 113/71 (Germany) and FRSF grant no. 01.07/071 (Ukraine).
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Mellit, A.S., Samoilenko, Y.S. & Vlasenko, M.A. Algebras Generated by Linearly Dependent Elements with Prescribed Spectra. Funct Anal Its Appl 39, 175–186 (2005). https://doi.org/10.1007/s10688-005-0036-2
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DOI: https://doi.org/10.1007/s10688-005-0036-2