Skip to main content
SpringerLink
Log in

Clifford semialgebras

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

Continuing the theory of systems, we introduce a theory of Clifford semialgebra systems, with application to representation theory via Hasse-Schmidt derivations on exterior semialgebras. Our main result, after the construction of the Clifford semialgebra, is a formula describing the exterior semialgebra as a representation of the Clifford semialgebra, given by the endomorphisms of the first wedge power.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Technically we are dealing with congruences, so the notation, which we use repeatedly, means that we are modding out the congruence generated by all \((x^j,0),\ j \ge n.\)

References

  1. Rowen, L.H.: Algebras with a negation map. Eur. J. Math. (2021). https://doi.org/10.1007/s40879-021-00499-0

    Article  MATH  Google Scholar 

  2. Behzad, O., Contiero, A., Gatto, L., Vidal Martins, R.: Polynomial ring representations of endomorphisms of exterior powers. Collectanea Math. 73, 107–133. (2022). https://doi.org/10.1007/s13348-020-00310-5

    Article  MathSciNet  MATH  Google Scholar 

  3. Behzad, O., Gatto, L.: Bosonic and fermionic representation of endomorphisms of exterior algebras, Fundamenta Mathematica. 256, 307–331 (2022), arXiv:2009.00479

  4. Gatto, L., Rowen, L.: Grassman semialgebras and the Cayley-Hamilton theorem, Proc. Amer. Math. Soc. series B, 7, 183–201 (2020), arXiv:1803.08093v1

  5. Giansirancusa, J., Giansirancusa, N.: A Grassmann algebra for matroids, Manuscripta Math. 156(1-2):187–213 (2018), arXiv:1510.04584v1 (2015)

  6. Jun, J., Rowen, L.: Categories with negation, in Categorical, homological and combinatorial methods in algebra, 221–270, Contemp. Math., 751, Amer. Math. Soc., [Providence], RI, (2020), arXiv:1709.0318

  7. Rowen, L.H.: An informal overview of triples and systems, Rings, modules and codes, 317–335, Contemp. Math., 727, Amer. Math. Soc., [Providence], RI, (2019)

  8. Akian, M., Gaubert, S., Rowen, L.: Examples of systems, preprint (2021)

  9. Jun, J., Mincheva, K., Rowen, L.: Homology of module systems. J. Pure Appl. Algebra 224(5), 106–243 (2020)

    Article  Google Scholar 

  10. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. III. Operator approach to the Kadomtsev–Petviashvili equation. J. Phys. Soc. Jpn. 50(11), 3806–3812 (1981). https://doi.org/10.1143/JPSJ.50.3806

    Article  MathSciNet  MATH  Google Scholar 

  11. Kac, V.G., Raina, A.K., Rozhkovskaya, N.: Bombay lectures on highest weight representations of infinite dimensional Lie algebras, second ed., Advanced Series in Mathematical Physics, vol. 29, World Scientific, (2013)

  12. Gatto, L., Salehyan, P.: Schubert derivations on the infinite wedge power. Bull. Braz. Math. Soc. 52(1), 149–174 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Laksov, D., Thorup, A.: A determinantal formula for the exterior powers of the polynomial ring. Indiana Univ. Math. J. 56(2), 825–845 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  15. Gatto, L., Salehyan, P: Hasse-Schmidt derivations on Grassmann algebras, IMPA monographs 4, Springer (2016)

  16. Gatto, L., Scherbak, I.: Cayley-Hamilton theorem in exterior algebra, functional analysis and geometry: Selim Grigorievich Krein centennial, 149–165, Contemp. Math., 733, Amer. Math. Soc., [Providence], RI, (2019)

  17. Bahadorykhalily, F.: Multivariate Hasse–Schmidt derivation on exterior algebras, Linear and Multilinear Algebra, https://doi.org/10.1080/03081087.2020.1809620

  18. Gatto, L., Salehyan, P.: The cohomology of the grassmannian is a \(gl_n\)-module. Commun. Algebra 48, 274–290 (2020). https://doi.org/10.1080/00927872.2019.1640240.. arXiv:1902.03824.pdf

    Article  MATH  Google Scholar 

  19. Behzad, O., Nasrollah, A.: Universal factorisation algebras of polynomials represent lie lgebras of endomorpisms,, J. Algebra and its applications, (2021), https://doi.org/10.1142/S0219498822500724; Available at (ArXiv:2006.07893.pdf.)

  20. Golan, J.: Semirings and their Applications, Springer, Dordrecht, (1999). (Previously published by Kluwer Acad. Publ., 1999.)

  21. Katsov, Y.: Tensor products and injective envelopes of semimodules over additively regular semirings. Algebra Colloquium 4(2), 121–131 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Izhakian, Z., Knebusch, M., Rowen, L.: Supertropical quadratic forms I. J. Pure Appl. Algebra 220(1), 61–93 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Izhakian, Z., Knebusch, M., Rowen, L.: Supertropical linear algebra. Pacific J. Math. 266(1), 43–75 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Skyrme, T.H.R.: Kinks and the Dirac equation. J. Math. Phys. 12, 1735–1743 (1971)

    Article  MathSciNet  Google Scholar 

  25. Behzad, O., Contiero, A., Martins, D.: On the vertex operator representation of Lie algebras of matrices, J. Algebra 597, 47–74 (2022)

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, Academic College of Tel-Aviv-Yaffo, Rabenu Yeruham St., P.O.B 8401, 6818211, Yaffo, Israel

    Adam Chapman

  2. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Letterio Gatto

  3. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Louis Rowen

Authors
  1. Adam Chapman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Letterio Gatto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Louis Rowen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louis Rowen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported by the Israel Science Foundation grant 1623/16. The research of the second author was supported by Finanziamento di Base della Ricerca,(no. 53_RBA17GATLET) and by INDAM–GNSAGA. The research of the third author was supported by Israel Science Foundation Grant No. 1623/16.

The authors thank the referee for careful readings, and for sound advice on improving the presentation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chapman, A., Gatto, L. & Rowen, L. Clifford semialgebras. Rend. Circ. Mat. Palermo, II. Ser 72, 1197–1238 (2023). https://doi.org/10.1007/s12215-022-00719-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-022-00719-w

Keywords

Mathematics Subject Classification

Search

Navigation