Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras

TL;DR

This paper extends a theorem on Lie algebra decomposition, showing that certain affine Kac-Moody algebras can be decomposed into orthogonal Heisenberg subalgebras, with detailed analysis for specific types.

Contribution

It introduces an affinization of a classical decomposition theorem, demonstrating new decompositions of affine Kac-Moody algebras into orthogonal Heisenberg subalgebras.

Findings

Decomposition of affine Kac-Moody algebras into orthogonal Heisenberg subalgebras.

Detailed analysis of types A_{p^m-1} and G_2.

Discussion of potential applications of these decompositions.

Abstract

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.