Rocky Mountain Journal of Mathematics

On the structure of split involutive Lie algebras

Antonio J. Calderón Martín and José M. Sánchez Delgado

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Abstract

We study the structure of arbitrary split involutive Lie algebras. We show that any of such algebras $L$ is of the form $L={\mathcal U} +\sum_{j}I_{j}$ with ${\mathcal U}$ a subspace of the involutive abelian Lie subalgebra $H$ and any $I_{j}$ a well described involutive ideal of $L$ satisfying $[I_j,I_k]=0$ if $j\neq k$. Under certain conditions, the simplicity of $L$ is characterized and it is shown that $L$ is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive Lie algebra.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 5 (2014), 1445-1455.

Dates
First available in Project Euclid: 1 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1420071549

Digital Object Identifier
doi:10.1216/RMJ-2014-44-5-1445

Mathematical Reviews number (MathSciNet)
MR3295637

Zentralblatt MATH identifier
1343.17015

Subjects
Primary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 17B20: Simple, semisimple, reductive (super)algebras 17B05: Structure theory

Keywords
Infinite dimensional Lie algebras involutive Lie algebras split Lie algebras roots root spaces

Citation

Martín, Antonio J. Calderón; Delgado, José M. Sánchez. On the structure of split involutive Lie algebras. Rocky Mountain J. Math. 44 (2014), no. 5, 1445--1455. doi:10.1216/RMJ-2014-44-5-1445. https://projecteuclid.org/euclid.rmjm/1420071549


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References

  • A.J. Calderón, On involutive Lie algebras having a Cartan decomposition, Bull. Austr. Math. Soc. 69 (2004), 191–202.
  • J.A. Cuenca, A. García and C. Martí n, Structure theory for $L^{*}$-algebras, Math-Proc. Camb. Phil. Soc. 107 (1990), 361–365.
  • B. Hsie, On complete Lie algebras, Comm. Alg. 34 (2006), 3743–-3750.
  • J.R. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc. 95 (1960), 69–80.
  • ––––, Cartan decompositions for $L^{*}$-algebras, Trans. Amer. Math. Soc. 98 (1961), 334–349.
  • N. Stumme, The structure of locally finite split Lie algebras, J. Alg. 220 (1999), 664–693.