Standard and Non-standard Extensions of Lie algebras

Abstract

We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root ${\alpha }_{+4}$, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if ${\alpha }_{+4}$ is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply-laced triple extended Lie algebras, obtaining all the non-simply laced ones. Non- standard extension procedures for a class of Lie algebras are proposed. It is shown that the 2-extensions of $E_8$, with a dot simply linked to the Dynkin-Kac diagram of $E_9$, are rank 10 subalgebras of $E_{10}$. Finally the simple root systems of a set of rank 11 subalgebras of $E_{11}$, containing as sub-algebra $E_{10}$, are explicitly written.

arXiv commentRevised version. Inaccurate statements corrected. Expanded version with added references

References in corpus (11)

• E_11 and M Theory
• Algebras, BPS States, and Strings
• $E_{11}$, SL(32) and Central Charges
• Very-extended Kac-Moody algebras and their interpretation at low levels
• Kac-Moody Symmetries of IIB Supergravity
• The Symmetry of M-Theories
• A class of Lorentzian Kac-Moody algebras
• Einstein billiards and overextensions of finite-dimensional simple Lie algebras
• Coset Symmetries in Dimensionally Reduced Bosonic String Theory
• Kac-Moody Symmetries of Ten-dimensional Non-maximal Supergravity Theories
• Remarks on E11 approach

Cited by in corpus (1)

• Spacelike Singularities and Hidden Symmetries of Gravity