Tokyo Journal of Mathematics

Algebraic Structure of the Lorentz and of the Poincaré Lie Algebras

Pablo ALBERCA BJERREGAARD, Dolores MARTÍN BARQUERO, Cándido MARTÍN GONZÁLEZ, and Daouda NDOYE

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We start with the Lorentz algebra $\text{\large$\mathfrak{L}$}=\text{\large$\mathfrak{o}$}_{\mathbb{R}}(1,3)$ over the reals and find a suitable basis $B$ such that the structure constants relative to it are integers. Thus we consider the $\mathbb{Z}$-algebra $\text{\large$\mathfrak{L}$}_{\mathbb{Z}}$ which is free as a $\mathbb{Z}$-module of which $B$ is $\mathbb{Z}$-basis. This allows us to define the Lorentz type algebra $\text{\large$\mathfrak{L}$}_K:=\text{\large$\mathfrak{L}$}_{\mathbb{Z}}\otimes_{\mathbb{Z}} K$ over any field $K$. In a similar way, we consider Poincaré type algebras over any field $K$. In this paper we study the ideal structure of Lorentz and of Poincaré type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground field has no square root of $-1$. Thus, they are simple over the reals but not over the complex. Also, if the ground field is of characteristic $2$ then Lorentz and Poincaré type algebras are neither simple nor semisimple. We extend the study of simplicity of the Lorentz algebra to the case of a ring of scalars where we have to use the notion of $\text{\large$\mathfrak{m}$}$-simplicity (relative to a maximal ideal $\text{\large$\mathfrak{m}$}$ of the ground ring of scalars). The Lorentz type algebras over a finite field $\mathbb{F}_q$ where $q=p^n$ and $p$ is odd are simple if and only if $n$ is odd and $p$ of the form $p=4k+3$. In case $p=2$ then the Lorentz type algebras are not simple. Once we know the ideal structure of the algebras, we get some information of their automorphism groups. For the Lorentz type algebras (except in the case of characteristic 2) we describe the affine group scheme of automorphisms and the derivation algebras. For the Poincaré algebras we restrict this program to the case of an algebraically closed field of characteristic other than 2.

Article information

Source
Tokyo J. Math., Volume 41, Number 2 (2018), 305-346.

Dates
First available in Project Euclid: 29 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1540800031

Mathematical Reviews number (MathSciNet)
MR3908798

Zentralblatt MATH identifier
07053480

Subjects
Primary: 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
Secondary: 17B20: Simple, semisimple, reductive (super)algebras 17B40: Automorphisms, derivations, other operators

Citation

ALBERCA BJERREGAARD, Pablo; MARTÍN BARQUERO, Dolores; MARTÍN GONZÁLEZ, Cándido; NDOYE, Daouda. Algebraic Structure of the Lorentz and of the Poincaré Lie Algebras. Tokyo J. Math. 41 (2018), no. 2, 305--346. https://projecteuclid.org/euclid.tjm/1540800031


Export citation

References

  • \BibAuthorsP. Alberca Bjerregaard, A. Elduque, C. Martín González and F. J. Navarro Márquez, On the Cartan-Jacobson Theorem, Journal of Algebra 250 (2002), Issue 2, 397–407.
  • \BibAuthorsP. Alberca Bjerregaard, O. Loos and C. Martín González, Derivations and automorphisms of Jordan algebras in characteristic two, Journal of Algebra 285 (2005), Issue 1, 146–181.
  • \BibAuthorsM. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
  • \BibAuthorsE. G. Beltrametti and A. A. Blasi, Dirac spinors, covariant currents and the Lorentz group over a finite field, Nuovo Cimento A 55 (1968), 301.
  • \BibAuthorsE. G. Beltrametti and A. A. Blasi, Rotation and Lorentz groups in a finite geometry, J. Math. Phys. 9 (1968), 1027–1035.
  • \BibAuthorsG. Benkart and E. Neher, The centroid of extended affine and root graded Lie algebras, Journal of Pure and Applied Algebra 205 (2006), Issue 1, 117–145.
  • \BibAuthorsH. Chandra, On the radical of a Lie algebra, Proceedings of the American Mathematical Society 1 (1950), 1, 14–17.
  • \BibAuthorsH. R. Coish, Elementary particles in a finite world geometry, Phys. Rev. 114 (1959), 383–388.
  • \BibAuthorsL. E. Dickson, Determination of the structure of all linear homogeneous groups in a Galois field which are defined by a quadratic invariant, Am. J. Math. 21 (1899), 193–256.
  • \BibAuthorsA. S. Dzhumadil'daev, On the cohomology of modular Lie algebras, Math. USSR Sb. 47 (1984), 127–143.
  • \BibAuthorsA. Elduque and M. Kochetov, Gradings on Simple Lie Algebras, Mathematical Surveys and Monographs, Providence, R.I. American Mathematical Society (AMS), 189, 2013.
  • \BibAuthorsS. Foldes, The Lorentz group and its finite field analogs: Local isomorphism and approximation, J. Math. Phys. 49 (2008), 093512.
  • \BibAuthorsG. M. D. Hogeweij, Almost-classical Lie algebras. I, Indagationes Mathematicae (Proceedings) 85 (1984), Issue 4,441–452.
  • \BibAuthorsJ. Hurley, Ideals in Chevalley algebras, Trans. Amer. Math. Soc. 137 (1969), 245–258.
  • \BibAuthorsN. Jacobson, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979.
  • \BibAuthorsM. A. Knus, A. Merkurjev, M. Rost and J. P. Tignol, The Book of Involutions. American Mathematical Society, Colloquium Publications vol. 44, 1998.
  • \BibAuthorsJ. S. Milne, Basic Theory of Affine Group Schemes 2012. http://www.jmilne.org/math/CourseNotes/AGS.pdf
  • \BibAuthorsE. H. Spanier, Algebraic topology, Corrected reprint, Springer-Verlag, New York-Berlin, 1981.
  • \BibAuthorsR. Steinberg, Automorphisms of classical Lie algebras, Pacific J. Math. 11 (1961), no. 3, 1119–1129. http://projecteuclid.org/euclid.pjm/1103037143
  • \BibAuthorsW. C. Waterhouse, Introduction to affine group schemes, Springer-Verlag, 1979.