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Algebras of general type: Rational parametrization and normal forms

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Abstract

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) 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 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∈I and {b j }j∈J, such that the ideal generated by the set {f i }i∈I is prime and, for every tuple c of structure constants satisfying the property b j (c) ≠ 0 for all j ∈ 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 ∈ I.

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References

  1. E. M. Andreev and V. L. Popov, “Stationary subgroups of points of general position in the representation space of a semisimple Lie group,” Funkts. Anal. Prilozh. 5 (4), 1–8 (1971) [Funct. Anal. Appl. 5, 265–271 (1971)].

    MathSciNet  Google Scholar 

  2. A. Borel, Linear Algebraic Groups, 2nd ed. (Springer, New York, 1991), Grad. Texts Math. 126.

    Book  MATH  Google Scholar 

  3. D. Boularas, “A new classification of planar homogeneous quadratic systems,” Qual. Theory Dyn. Syst. 2, 93–110 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  4. N. Bourbaki, Algebra I. Chapters 1–3 (Hermann, Paris, 1974), Elements of Mathematics.

    MATH  Google Scholar 

  5. N. Bourbaki, Algebra II. Chapters 4–7 (Springer, Berlin, 1990), Elements of Mathematics.

    MATH  Google Scholar 

  6. R. Durán Daz, J. Mu˜noz Masqué, and A. Peinado Domnguez, “Classifying quadratic maps from plane to plane,” Linear Algebra Appl. 364, 1–12 (2003).

    Article  MathSciNet  Google Scholar 

  7. W. Fulton and J. Harris, Representation Theory: A First Course (Springer, New York, 1991), Grad. Texts Math. 129.

    MATH  Google Scholar 

  8. D. G. Higman, “Indecomposable representations at characteristic p,” Duke Math. J. 21, 377–381 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Mu˜noz Masqué and M. E. Rosado Mara, “Rational invariants on the space of all structures of algebras on a two-dimensional vector space,” Electron. J. Linear Algebra 23, 483–507 (2012).

    MathSciNet  Google Scholar 

  10. V. L. Popov, “An analogue of M. Artin’s conjecture on invariants for nonassociative algebras,” in Lie Groups and Lie Algebras: E.B. Dynkin’s Seminar (Am. Math. Soc., Providence, RI, 1995), AMS Transl., Ser. 2, 169, pp. 121–143.

    Google Scholar 

  11. V. L. Popov, “Rationality and the FML invariant,” J. Ramanujan Math. Soc. 28A (Spec. Iss.), 409–415 (2013).

    MathSciNet  MATH  Google Scholar 

  12. V. L. Popov and E. B. Vinberg, “Invariant theory,” in Algebraic Geometry IV (Springer, Berlin, 1994), Encycl. Math. Sci. 55, pp. 123–278.

    Chapter  Google Scholar 

  13. T. Ralley, “Decomposition of products of modular representations,” J. London Math. Soc. 44, 480–484 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  14. Z. Reichstein and A. Vistoli, “Birational isomorphisms between twisted group actions,” J. Lie Theory 16 (4), 791–802 (2006).

    MathSciNet  MATH  Google Scholar 

  15. M. Rosenlicht, “Some basic theorems on algebraic groups,” Am. J. Math. 78, 401–443 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  16. D. J. Saltman, “Noether’s problem over an algebraically closed field,” Invent. Math. 77, 71–84 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. D. Schafer, An Introduction to Nonassociative Algebras (Academic, New York, 1966).

    MATH  Google Scholar 

  18. J.-P. Serre, “Espaces fibrés algébriques,” in Anneaux de Chow et applications: Sémin. C. Chevalley, 1958 (Secr. Math., Paris, 1958), Exp. 1, pp. 1–37.

    Google Scholar 

  19. A. Speiser, “Zahlentheoretische Sätze aus der Gruppentheorie,” Math. Z. 5, 1–6 (1919).

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Steinberg, Conjugacy Classes in Algebraic Groups (Springer, Berlin, 1974), Lect. Notes Math. 366.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Vladimir L. Popov

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  1. Vladimir L. Popov
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Correspondence to Vladimir L. Popov.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 292, pp. 209–223.

To V.P. Platonov on his 75th anniversary

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Popov, V.L. Algebras of general type: Rational parametrization and normal forms. Proc. Steklov Inst. Math. 292, 202–215 (2016). https://doi.org/10.1134/S0081543816010132

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  • DOI: https://doi.org/10.1134/S0081543816010132

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