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.
Similar content being viewed by others
Notes
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
Rowen, L.H.: Algebras with a negation map. Eur. J. Math. (2021). https://doi.org/10.1007/s40879-021-00499-0
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
Behzad, O., Gatto, L.: Bosonic and fermionic representation of endomorphisms of exterior algebras, Fundamenta Mathematica. 256, 307–331 (2022), arXiv:2009.00479
Gatto, L., Rowen, L.: Grassman semialgebras and the Cayley-Hamilton theorem, Proc. Amer. Math. Soc. series B, 7, 183–201 (2020), arXiv:1803.08093v1
Giansirancusa, J., Giansirancusa, N.: A Grassmann algebra for matroids, Manuscripta Math. 156(1-2):187–213 (2018), arXiv:1510.04584v1 (2015)
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
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)
Akian, M., Gaubert, S., Rowen, L.: Examples of systems, preprint (2021)
Jun, J., Mincheva, K., Rowen, L.: Homology of module systems. J. Pure Appl. Algebra 224(5), 106–243 (2020)
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
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)
Gatto, L., Salehyan, P.: Schubert derivations on the infinite wedge power. Bull. Braz. Math. Soc. 52(1), 149–174 (2021)
Laksov, D., Thorup, A.: A determinantal formula for the exterior powers of the polynomial ring. Indiana Univ. Math. J. 56(2), 825–845 (2007)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)
Gatto, L., Salehyan, P: Hasse-Schmidt derivations on Grassmann algebras, IMPA monographs 4, Springer (2016)
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)
Bahadorykhalily, F.: Multivariate Hasse–Schmidt derivation on exterior algebras, Linear and Multilinear Algebra, https://doi.org/10.1080/03081087.2020.1809620
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
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.)
Golan, J.: Semirings and their Applications, Springer, Dordrecht, (1999). (Previously published by Kluwer Acad. Publ., 1999.)
Katsov, Y.: Tensor products and injective envelopes of semimodules over additively regular semirings. Algebra Colloquium 4(2), 121–131 (1997)
Izhakian, Z., Knebusch, M., Rowen, L.: Supertropical quadratic forms I. J. Pure Appl. Algebra 220(1), 61–93 (2016)
Izhakian, Z., Knebusch, M., Rowen, L.: Supertropical linear algebra. Pacific J. Math. 266(1), 43–75 (2013)
Skyrme, T.H.R.: Kinks and the Dirac equation. J. Math. Phys. 12, 1735–1743 (1971)
Behzad, O., Contiero, A., Martins, D.: On the vertex operator representation of Lie algebras of matrices, J. Algebra 597, 47–74 (2022)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-022-00719-w
Keywords
- Clifford semialgebras
- Exterior semialgebras
- Schubert derivations
- Exterior semialgebra representation of endomorphisms
- Bosonic vertex operator representation of Lie semialgebras of endomorphisms