Lie Algebra Upper Central Series

The Lie algebra upper central series of a Lie algebra is an ascending sequence of subalgebras constructed by taking centers of successive quotients (de Ruiter 1972). It is also called the upper central sequence (Hrivnák and Novotný 2004). Starting with , define

(1)

where consists of the elements for which the Lie bracket vanishes for every . Equivalently,

(2)

Each term is an ideal of , meaning a subspace satisfying . Here, denotes the vector space span of the Lie brackets with and .

The Lie algebra is a nilpotent Lie algebra iff for some nonnegative integer . The least such is its nilpotency class, which also equals the number of steps needed for the Lie algebra lower central series to reach 0.

For example, let have vector space basis , , and , with and . This is the three-dimensional Lie algebra associated with the Heisenberg group. Its Lie algebra upper central series is

(3)

where denotes the vector space span of , so has nilpotency class 2.