A Lie algebra ℒ is said to be
ad-associative if the image of the adjoint representation of ℒ on ℒ is an associative
algebra under composition. We show that every ad-associative Lie algebra is a
quotient of a left commutative (xyw = yxw) associative algebra by a Lie ideal. We
conclude that every ad-associative Lie algebra is solvable and every irreducible
representation of a nilpotent, ad-associative Lie group is square integrable modulo its
kernel. We also characterize the HAT algebras of Howe [2] in terms of associative
algebra.
