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A generalization of Cayley–Hamilton algebras and an introduction to their geometries

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Abstract

Let A and B be graded algebras in the same variety of trace algebras, such that A is a finite-dimensional, central simple power associative algebra (in the ordinary sense). Over a field K of characteristic zero, we study sufficient conditions that ensure B to be a graded subalgebra of A. More precisely, we prove, under additional hypotheses, that there is a graded and trace-preserving embedding from B to A over some associative and commutative K-algebra C if and only if B satisfies all G-trace identities of A over K. As a consequence of these results, we give a geometric interpretation of our main theorem under the context of graded algebras, and we apply them beyond the Cayley–Hamilton algebras presented in [24, 29]. Such results open a wide range of opportunities to study geometry in Jordan and alternative algebras (with trivial grading).

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References

  1. E. Aljadeff, A. Kanel-Belov and Y. Karasik, Kemer’s theorem for affine PI algebras over a field of characteristic zero, Journal of Pure and Applied Algebra 220 (2016), 2771–2808.

    Article  MathSciNet  Google Scholar 

  2. S. A. Amitsur, A noncommutative Hilbert basis theorem and subrings of matrices, Transactions of the American Mathematical Society 149 (1970), 133–142.

    Article  MathSciNet  Google Scholar 

  3. Y. Bahturin and M. Kochetov, Classification of group gradings on simple Lie algebra of types \({\cal A},{\cal B},{\cal C}\) and \({\cal D}\), Journal of Algebra 324 (2010), 2971–2989.

    Article  MathSciNet  Google Scholar 

  4. A. Berele, Trace identities and ℤ/2ℤ-graded invariants, Transactions of the American Mathematical Society 309 (1988), 581–589.

    MathSciNet  Google Scholar 

  5. A. Berele, Matrices with involution and invariant theory, Journal of Algebra 135 (1990), 139–164.

    Article  MathSciNet  Google Scholar 

  6. A. Berele, Embedding theorems for superalgebras, Communications in Algebra 21 (1993), 4179–4182.

    Article  MathSciNet  Google Scholar 

  7. A. Berele, Supertraces and matrices over Grassmann algebras, Advances in Mathematics 108 (1994), 77–90.

    Article  MathSciNet  Google Scholar 

  8. A. Berele, Examples and counterexamples for M1,1 embeddings, Journal of Algebra 172 (1995), 379–384.

    Article  MathSciNet  Google Scholar 

  9. L. A. Bokut, Imbedding of rings, Uspekhi Matematicheskikh Nauk 42 (256) (1987), 87–111; English Translation: Russian Mathematical Surveys 42 (1987), 105–138.

    MathSciNet  Google Scholar 

  10. V. Drensky, Free Algebras and PI-algebras, Springer, Singapore, 1999.

    Google Scholar 

  11. V. Drensky and E. Formanek, Polynomial identity rings, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser, Basel, 2004.

    Google Scholar 

  12. A. Elduque and M. Kochetov, Gradings on Simple Lie Algebras, Mathematical Surveys and Monographs, Vol. 189, American Mathematical Society, Providence, RI, 2013.

    Google Scholar 

  13. C. Fidelis, D. Diniz and P. Koshlukov, Embeddings for the Jordan algebra of a bilinear form, Advances in Mathematics 337 (2018), 294–316.

    Article  MathSciNet  Google Scholar 

  14. C. Fidelis, P. Koshlukov and I. Shestakov, Trace identities for special Jordan algebras, submitted.

  15. A. de França and I. Sviridova, Köthe’s problem, Kurosh–Levitzky problem and graded rings, Journal of Algebra 602 (2022), 224–246.

    Article  MathSciNet  Google Scholar 

  16. A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, Vol. 122, American Mathematical Society, Providence, RI, 2005.

    Google Scholar 

  17. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer, New York–Heidelberg, 1977.

    Google Scholar 

  18. A. Iltyakov, Laplace operator and polynomial invariants, Journal of Algebra 207 (1998), 256–271.

    Article  MathSciNet  Google Scholar 

  19. N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, RI, 1964.

    Google Scholar 

  20. N. Jacobson, Structure and Representation of Jordan Algebras, Vol. 39, American Mathematical Society, Providence, RI, 1968.

    Book  Google Scholar 

  21. A. Kanel-Belov and L. H. Rowen, Computational Aspects of Polynomial Identities, Research Notes in Mathematics, Vol. 9, A K Peters, Wellesley, MA, 2005.

    Book  Google Scholar 

  22. A. R. Kemer, Ideals of identities of associative algebras, Translations of Mathematical Monographs, Vol. 87, American Mathematical Society, Providence, RI, 1991.

    Google Scholar 

  23. E. N. Kuzmin, On the Nagata–Higman theorem, in Mathematical Structures, Computational Mathematics, Mathematical Modelling, Sofia, 1975, pp. 101–107.

  24. L. Le Bruyn, Noncommutative Geometry and Cayley-smooth Orders, Pure and Applied Mathematics (Boca Raton), Vol. 290, Chapman & Hall/CRC, Boca Raton, FL, 2008.

    Google Scholar 

  25. A. I. Mal’tsev, On the immersion of an algebraic ring into a field, Mathematische Annalen 113 (1937), 686–691.

    Article  MathSciNet  Google Scholar 

  26. A. I. Mal’tsev, On the inclusion of associative systems in groups, Matematicheskii Sbornik 6 (1939), 331–336.

    Google Scholar 

  27. A. I. Mal’tsev, On the inclusion of associative systems in groups II, Matematicheskii Sbornik 8 (1940), 251–264.

    Google Scholar 

  28. C. Procesi, The invariant theory of n × n matrices, Advances in Mathematics 19 (1976), 306–381.

    Article  MathSciNet  Google Scholar 

  29. C. Procesi, A formal inverse to the Cayley–Hamilton theorem, Journal of Algebra 107 (1987), 63–74.

    Article  MathSciNet  Google Scholar 

  30. Yu. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic 0, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 38 (1974), 723–756; English Translation: Mathematics of the USSR - Izvestiya 8 (1975), 724–760.

    MathSciNet  Google Scholar 

  31. W. F. Santos and A. Rittatore, Actions and Invariants of Algebraic Groups, Pure and Applied Mathematics (Boca Raton), Vol. 269, Chapman & Hall/CRC, Boca Raton, FL, 2005.

    Book  Google Scholar 

  32. G. W. Schwarz, Invariant theory of G2, Bulletin of the American Mathematical Society 9 (1983), 335–338.

    Article  Google Scholar 

  33. G. W. Schwarz, Invariant theory of G2 and Spin7, Commentarii Mathematici Helvetici 63 (1988), 624–663.

    Article  MathSciNet  Google Scholar 

  34. L. W. Small, An example in PI-rings, Journal of Algebra 17 (3) (1971), 434–436.

    Article  MathSciNet  Google Scholar 

  35. T. A. Springer and F. D. Veldkamp, in Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Springer, Berlin, 2000.

    Book  Google Scholar 

  36. S. Vasilovskii, Trace identities of the Jordan algebra of a bilinear form, Siberian Mathematical Journal 32 (1991), 916–919.

    Article  MathSciNet  Google Scholar 

  37. C. Wall, Graded Brauer groups, Journal für die Reine und Angewandte Mathematik 213 (1963) 187–199.

    MathSciNet  Google Scholar 

  38. H. Weyl, The classical Groups, Princeton University Press, Princeton, NJ, 1946.

    Google Scholar 

  39. K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov, Rings That Are Nearly Associative, Pure and Applied Mathematics, Vol. 104, Academic Press, New York–London, 1982.

    Google Scholar 

  40. A. N. Zubkov and I. P. Shestakov, Invariants of G2 and Spin(7) in positive characteristic, Transformation Groups 23 (2018), 555–588.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Many thanks to the anonymous referee for spotting several inaccuracies and helping to improve the text.

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Author notes

  1. Claudemir Fidelis

    Present address: Department of Mathematics, Universidade Estadual de Campinas, Sergio Buarque de Holanda, 651, IMECC, UNICAMP, 13083-859, Campinas, SP, Brazil

Authors and Affiliations

  1. Department of Mathematics, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, 30123-970, Belo Horizonte, MG, Brazil

    Charles Almeida

  2. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Av. Aprígio Veloso 785, 58109-970, Campina Grande, PB, Brazil

    Claudemir Fidelis & José Lucas Galdino

Authors
  1. Charles Almeida
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  2. Claudemir Fidelis
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  3. José Lucas Galdino
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Corresponding author

Correspondence to Claudemir Fidelis.

Additional information

Partially suported by FAPEMIG Universal project number APQ-01655-22, CNPQ project number 408974/2023-0, and FAPESQ-PB grant No. 2021/3176.

Supported by CNPq grant No. 305651/2021-8 and FAPESP grant No. 2023/04011-9.

Partially supported by CNPq grant No. 404851/2021-5 and by a PhD grant from CAPES, Brazil.

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Almeida, C., Fidelis, C. & Galdino, J.L. A generalization of Cayley–Hamilton algebras and an introduction to their geometries. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2614-0

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  • DOI: https://doi.org/10.1007/s11856-024-2614-0

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