Um convite à teoria algébrica de formas quadráticas e de formas hermitianas

Kaique Santos; Hugo  Luiz Mariano

doi:10.14244/lajm.v1i01.9

Authors

Kaique Santos IME-USP
Hugo Luiz Mariano

DOI:

https://doi.org/10.14244/lajm.v1i01.9

Keywords:

quadratic forms, Witt ring, algebras with involution, hermitian forms

Abstract

The purpose of this work is twofold: to provide an introduction for readers unfamiliar with the algebraic theory of forms quadratics over fields (ATQF) and, subsequently, with the referent constructed in the ATQF, to introduce the algebraic theory of hermitian forms with coefficients in associative algebras equipped with an involution (ATHF).

References

Lam T. Y. Introduction to quadratic forms over fields. vol. 67 of Graduate Studies in Mathematics. 1st ed. American Mathematical Society; 2005. DOI: https://doi.org/10.1090/gsm/067/01

Santos D. F. Elementos da teoria algébrica de formas quadráticas e de seus aneis graduados; 2015.

Roberto K. M. A. Multi-anéis e a Câmara Secreta: relações funtoriais entre teorias abstratas de Formas Quadráticas; 2019.

Pfister A. Multiplikative Quadratische Formen. Archiv der Mathematik. 1965;16(1):363-70. DOI: https://doi.org/10.1007/BF01220043

Milnor J. Algebraic K-theory and quadratic forms. Inventiones Mathematicae. 1970;9(4):318-44. DOI: https://doi.org/10.1007/BF01425486

Lam T. Y. Orderings, valuations, and quadratic forms. vol. 52 of Regional Conference Series in Mathematics. 1st ed. American Mathematical Society; 1983. DOI: https://doi.org/10.1090/cbms/052

Mariano H.L., Ribeiro H. R.O. , Roberto K. M. A. Uma jornada pelas teorias algébricas de formas quadráticas. 1st ed. Livraria da Física-LF; 2021.

Witt E. Theorie der quadratischen Formen in beliebigen Körpern. Journal für die reine und angewandte Mathematik. 1937;176(1):31-44. DOI: https://doi.org/10.1515/crll.1937.176.31

Lam T. Y. Ten Lectures on Quadratic Forms over Fields. Queen’s Papers in Pure and Applied Mathematics. 1977;46(1):1-102.

Knus M. A. Quadratic and Hermitian forms over rings. vol. 294 of Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics. 1st ed. Springer-Verlag; 1991. DOI: https://doi.org/10.1007/978-3-642-75401-2_1

Scharlau W. Quadratic and Hermitian forms. vol. 270 of Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics. 1st ed. Springer-Verlag; 1985. DOI: https://doi.org/10.1007/978-3-642-69971-9_1

Astier V., Unger T. Signatures of hermitian forms and "prime ideals" of Witt groups. Advances in Mathematics. 2015;285(1):497-514. DOI: https://doi.org/10.1016/j.aim.2015.07.035

Bayer-Fluckiger E, Moldovan D. A. Sesquilinear forms over rings with involution. Journal of Pure and Applied Algebra. 2014;218(1):417-23. DOI: https://doi.org/10.1016/j.jpaa.2013.06.012

Astier V, Unger T. Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher. Journal of Algebra. 2018;508(1):339-63. DOI: https://doi.org/10.1016/j.jalgebra.2018.05.004

Astier V, Unger T. Signatures of hermitian forms and the Knebusch trace formula. Advances in Mathematics. 2014;358(1):925-47. DOI: https://doi.org/10.1007/s00208-013-0977-3

Astier V, Unger T. Signatures of hermitian forms and applications (preprint version). Oberwolfach Reports. 2013;31(1):1-3.

Astier V, Unger T. Signatures, sums of hermitian squares and positive cones on algebras with involution. Electronic Research Announcements in Mathematical Sciences. 2018;28(1):16-26. DOI: https://doi.org/10.3934/era.2018.25.003

Astier V, Unger T. Positive cones on algebras with involution. Advances in Mathematics. 2020;361(1):106954. DOI: https://doi.org/10.1016/j.aim.2019.106954

Astier V, Unger T. Positive Cones and Gauges on Algebras With Involution. International Mathematics Research Notices. 2022;2022(10):7259-303. DOI: https://doi.org/10.1093/imrn/rnaa337