On the non-abelian tensor product of Lie algebras

Abstract

Ellis (G. Ellis, A non-abelian tensor product of Lie algebras, Glasgow Math. J. 39 (1991), pp. 101–120.) introduced the notion of the non-abelian tensor product L ⊗ K for a pair of Lie algebras L, K and investigated some of its fundamental properties. In this article, we study some common properties between Lie algebras and their tensor products, and present some bounds on the nilpotency class and solvability length of L ⊗ K, provided such information is given on L or K. Also, we give some upper and lower bounds for the dimension of L ⊗ K if L and K are finite-dimensional nilpotent Lie algebras and ideals of a single Lie algebra.