This is the second in a series of papers dedicated to studying
w-knots, and more generally, w-knotted objects (w-braids, w-tangles,
etc.). These are classes of knotted objects that are \underline{w}ider but
\underline{w}eaker than their ``\underline{u}sual'' counterparts. To
get (say) w-knots from usual knots (or u-knots), one has to allow non-planar ``virtual''
knot diagrams, hence enlarging the the base set of knots. But then one
imposes a new relation beyond the ordinary collection of Reidemeister moves,
called the ``overcrossings commute'' relation, making w-knotted
objects a bit weaker once again.
Satoh~\cite{Satoh:RibbonTorusKnots} studied several classes of
w-knotted objects (under the name ``\underline{w}eakly-virtual'') and has
shown them to be closely related to certain classes of knotted surfaces
in $\bbR^4$.
In this article we study finite type invariants of w-tangles and
w-trivalent graphs (also referred to as w-tangled foams).
Much as the spaces $\calA$ of chord diagrams for ordinary knotted
objects are related to metrized Lie algebras, the spaces
$\calA^w$ of ``arrow diagrams'' for w-knotted objects are related to
not-necessarily-metrized Lie algebras. Many questions concerning w-knotted
objects turn out to be equivalent to questions about Lie algebras. Most
notably we find that a homomorphic universal finite type invariant of
w-foams is essentially the same as a solution of
the Kashiwara-Vergne~\cite{KashiwaraVergne:Conjecture} conjecture and
much of the Alekseev-Torossian~\cite{AlekseevTorossian:KashiwaraVergne}
work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted
as a study of w-foams.