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 Tr. Mat. Inst. Steklova, 2016, Volume 292, Pages 209–223 (Mi tm3700)

Algebras of general type: Rational parametrization and normal forms

V. L. Popov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Finite-dimensional (not necessarily associative) $\boldsymbol k$-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over $\boldsymbol k$) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, $\{f_i\}_{i\in I}$ and $\{b_j\}_{j\in J}$, such that the ideal generated by the set $\{f_i\}_{i\in I}$ is prime and, for every tuple $c$ of structure constants satisfying the property $b_j(c)\neq 0$ for all $j\in J$, there exists a unique new basis of this algebra in which the tuple $c'$ of its structure constants satisfies the property $f_i(c')=0$ for all $i\in I$.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968516010131

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 292, 202–215

Bibliographic databases:

ArXiv: 1411.6570
Document Type: Article
UDC: 512

Citation: V. L. Popov, “Algebras of general type: Rational parametrization and normal forms”, Algebra, geometry, and number theory, Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday, Tr. Mat. Inst. Steklova, 292, MAIK Nauka/Interperiodica, Moscow, 2016, 209–223; Proc. Steklov Inst. Math., 292 (2016), 202–215

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3700
• https://doi.org/10.1134/S0371968516010131
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This publication is cited in the following articles:
1. Vik. S. Kulikov, E. I. Shustin, “On $G$-Rigid Surfaces”, Proc. Steklov Inst. Math., 298 (2017), 133–151
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