Engel conditions and symmetric tensors

Abstract

In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Our main result generalizes this in the language of tensor algebra, and describes a relatively small generating set for the module generated by all n-th tensor powers of elements of a finitely generated ℤ-module M, in terms of a generating set for M.

Keywords:

• Engel condition
• Lie ring
• symmetrization
• tensor

AMS Subject Classifications::

• Primary 15A69
• secondary 17B01
• 20F45

Acknowledgements

The author is grateful to Michael Vaughan-Lee for stimulating discussions on 5, an extended version of this article. The author acknowledges partial support from MIUR-Italy via PRIN ‘Lie rings and algebras, groups, cryptography’.