The second phase is solved using formal algebraic computations. This
means that the particular vector fields, differentiation, manifolds,
and so on, can be ignored. The concepts involve pure algebraic
manipulation. To avoid confusion with previous definitions, the term
*formal* will be added to many coming definitions. Recall from
Section 4.4.1 the formal definitions of the algebra of
polynomials (e.g.,
). Let
denote the *formal
noncommutative algebra*^{15.11} of polynomials in the variables ,
, . The here are treated as symbols and have no
other assumed properties (e.g, they are not necessarily vector
fields). When polynomials are multiplied in this algebra, no
simplifications can be made based on commutativity. The algebra can
be converted into a Lie algebra by defining a Lie bracket. For any
two polynomials
, define the *formal
Lie bracket* to be
. The formal Lie bracket yields an
equivalence relation on the algebra; this results in a *formal Lie
algebra*
(there are many equivalent expressions
for the same elements of the algebra when the formal Lie bracket is
applied). Nilpotent versions of the formal algebra and formal Lie
algebra can be made by forcing all monomials of degree to be
zero. Let these be denoted by
and
, respectively. The P. Hall basis can be applied to obtain a basis of the formal Lie
algebra. Example 15.18 actually corresponds to the basis of
using formal calculations.

Steven M LaValle 2012-04-20