1.6. Inverse conjecture over the integers 99

(i.e., the group of upper-triangular unipotent matrices with arbitrary real

entries in the upper triangular positions) with

G≥2 :=

⎛

⎝0

1 0 R

1 0

0 0 1

⎞

⎠

and G≥i trivial for i 2 (so in this case, G≥i is also the lower central series).

Exercise 1.6.13. Show that a sequence

g(n) =

⎛

⎝0

1 x(n) y(n)

1

z(n)⎠

0 0 1

⎞

from Z to the Heisenberg group G is a polynomial sequence if and only if

x, z are linear polynomials and z is a quadratic polynomial.

It is a standard fact in the theory of Lie groups that a connected, simply

connected nilpotent Lie group G is topologically equivalent to its Lie algebra

g, with the homeomorphism given by the exponential map exp: g → G

(or its inverse, the logarithm function log: G → g. Indeed, the Baker-

Campbell-Hausdorff formula lets one use the nilpotent Lie algebra g to build

a connected, simply connected Lie group with that Lie algebra, which is

then necessarily isomorphic to G. One can thus classify filtered nilpotent

Lie groups in terms of filtered nilpotent Lie algebras, i.e., a nilpotent Lie

algebra g = g≥0 together with a nested family of sub-Lie algebras

g≥0 ≥ g≥1 ≥ · · · ≥ g≥s+1 = {0}

with the inclusions [gi, gj] ⊂ gi+j (in which the bracket is now the Lie bracket

rather than the commutator). One can describe such filtered nilpotent Lie

algebras even more precisely using Mal’cev bases; see [Ma1949], [Le2005].

For instance, in the case of the Heisenberg group, one has

g = g≥0 = g≥1 :=

⎛

⎝0

0 R R

0

R⎠

0 0 0

⎞

and

g≥2 :=

⎛

⎝0

0 0 R

0 0

0 0 0

⎞

⎠

.

From the filtration property, we see that for i ≥ 0, each G≥i+1 is a normal

closed subgroup of G≥i, and for i ≥ 1, the quotient group G≥i+1/G≥i is a

connected, simply connected abelian Lie group (with Lie algebra g≥i+1/g≥i),

and is thus isomorphic to a vector space (with the additive group law).

Related to this, one can view G = G≥0 as a group extension of the quotient