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Differential Calculuses on Group Algebras

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Abstract

Families of operators obeying some inductive identities (such as the Leibniz rule—the case of derivations and Fox derivations) as characters on a suitable groupoid are described. First and foremost, this construction is implemented for derivations in group algebras and Fox derivations as characters on an action groupoid.

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REFERENCES

  1. Yu. V. Matiyasevich, “Enumerable sets are diophantine,” Sov. Math., Dokl. 191, 354–358 (1970).

    MATH  Google Scholar 

  2. O. Kharlampovich and A. Myasnikov, “Equations in algebras,” Int. J. Algebra Comput. 28, 1517–1533 (2018). https://doi.org/10.1142/S0218196718400064

    Article  MathSciNet  MATH  Google Scholar 

  3. A. I. Perov and I. D. Kostrub, “Differential equations in Banach algebras,” Dokl. Math. 101, 139–143 (2020). https://doi.org/10.1134/s1064562420020179

    Article  MathSciNet  MATH  Google Scholar 

  4. A. I. Perov and I. D. Kostrub, “On differential equations in Banach algebras,” Vestn. Ross. Univ. Mat. 25, 410–421 (2020). https://doi.org/10.20310/2686-9667-2020-25-132-410-421

    Article  MATH  Google Scholar 

  5. A. N. Parshin, “On a ring of formal pseudodifferential operators,” Proc. Steklov Inst. Math. 224, 266–280 (1999).

    MATH  Google Scholar 

  6. A. A. Arutyunov, “On a pseudodifferentional calculation in group algebras,” Adv. Syst. Sci. Appl. 22 (4), 11–18 (2022). https://doi.org/10.25728/assa.2022.22.4.1342

    Article  Google Scholar 

  7. A. A. Arutyunov, A. S. Mishchenko, and A. I. Shtern, “Derivations of group algebras,” J. Math. Sci. 248, 709–718 (2016). https://doi.org/10.1007/s10958-020-04906-x

    Article  MathSciNet  MATH  Google Scholar 

  8. A. A. Arutyunov and A. S. Mishchenko, “A smooth version of Johnson’s problem on derivations of group algebras,” Sb. Math. 210, 756–782 (2019). https://doi.org/10.1070/sm9119

    Article  MathSciNet  MATH  Google Scholar 

  9. A. A. Arutyunov and A. V. Alekseev, “Complex of n-categories and derivations in group algebras,” Topology Its Appl. 275, 107002 (2020). https://doi.org/10.1016/j.topol.2019.107002

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Burghelea, “The cyclic homology of the group rings,” Comment. Mathematici Helv. 60, 354–365 (1985). https://doi.org/10.1007/bf02567420

    Article  MathSciNet  MATH  Google Scholar 

  11. A. S. Mischenko, “Derivations of group algebras and Hochschild cohomology,” in Differential Equations on Manifolds and Mathematical Physics, Ed. by V. M. Manuilov, A. S. Mishchenko, V. E. Nazaikinskii, B. W. Schulze, and W. Zhang, Trends in Mathematics (Birkhäuser, Cham, 2021), pp. 263–272. https://doi.org/10.1007/978-3-030-37326-9_16

  12. L. Creedon and K. Hughes, “Derivations on group algebras with coding theory applications,” Finite Fields Their Appl. 56, 247–265 (2019). https://doi.org/10.1016/j.ffa.2018.11.005

    Article  MathSciNet  MATH  Google Scholar 

  13. R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 57 (Springer, Berlin, 1963). https://doi.org/10.1007/978-1-4612-9935-6

  14. R. H. Fox, “Free differential calculus. I: Derivation in the free group ring,” Ann. Math. 57, 547–560 (1953). https://doi.org/10.2307/1969736

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation, project no. 20-11-20131.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 117997, Moscow, Russia

    A. A. Arutyunov

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  1. A. A. Arutyunov
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Correspondence to A. A. Arutyunov.

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Translated by V. Arutyunyan

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Arutyunov, A.A. Differential Calculuses on Group Algebras. Russ Math. 67, 77–81 (2023). https://doi.org/10.3103/S1066369X23060014

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