N-commutators for simple Lie algebras

Askar Serkulovich Dzhumadil'daev, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmeier, Theodore Voronov

2007

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DOI 10.1063/1.2820963
Data confidence 70 medium
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Abstract

The N‐commutator sN(X1,…,XN) of N vector fields X1,…,XN (differential operators of order 1) is defined as the skew‐symmetric sum of the N! compositions Xσ(1)⋯Xσ(N) for all permutations σ∈SymN. In general it is a differential operator of order more than 1, but for some special cases of N it might happen that sN will be a well‐defined operation on the space of vector fields. We construct such N‐commutators for simple Lie algebras for the four classical series and for the exceptional algebra G2. We establish that G2 has well‐defined 2‐ and 10‐commutators.