Jerzy,
I think Torsten is on vacation. I wrote this on his blog site a couple of days ago. It took me a while to get back to this. I have been working some on how Yangians work in this. The Chern-Simons Lagrangian and the appearance of knot crossing equations seems to suggest there is a dual gauge form of the Yang-Baxter equation underlying this.
A Lie algebra gl(N) is a set of NxN matrices with a bracket structure [A, B] that is in general nonzero. The matrices E_{ij} satisfy
[E_{ij}, E_{kl}] = δ_{kj}E_{il} - δ_{il}E_{kj}
A universal enveloping algebra (UEA) is constructed by finding the most general Lie algebra over a field that contains this Lie algebra. A common structure is
U(E) = E + ExE + ExExE + ... .
The q-deformation of the UEA is the Yangian Y(gl(N))with generators T^r_{ij}
[T^r_{ij}, T^s_{kl}] =
sum_{t=0}^{min(r,s)}T^s_{kl}T^{r+s-t-1}_{il} - T^{r+s-t-1}_{jk}T^t_{il}]
which recovers the UEA for r = s = 1 such that
[T^1_{ij}, T^1_{kl}] = δT^s_{kl}T^1_{il} - T^1_{jk}δ^1_{il}]
Now define a a power series in the generator T_{ij} = δ_{ij} + T^1_{ij}u^{-1} + T^2_{ij}u^{-2} + ... . The definition of the Yangian is obtains by substitution into this q-dformation of generators
(u - v)[T_{ij}(u), T_{kl}(v)] = T_{kj}(u)T_{il)(v) - T_{kj}(v)T_{il)(u}.
The double commutator (u - v)[[T_{ij}(u), T_{kl}(v)], T_{mn}(u+v)] results in
(u - v)[[T_{ij}(u), T_{kl}(v)], T_{mn}(u+v)]
= (1/u)T_{kj}(u)(T_{il}(v)T_{mn}(u+v) - T_{il}(u+v)T_{mn}(v))
- (1/v)(T_{kj}(u)T_{mn}(u+v) - T_{kj}(u+v)T_{mn}(v))T_{il}(v)
+ (1/v)T_{kj}(v)(T_{il}(v)T_{mn}(u+v) - T_{il}(u+v)T_{mn}(u))
+ (1/v)(T_{kj}(v)T_{mn}(u+v) - T_{kj}(u+v)T_{mn}(v))T_{il}(u))
which while rather complicated is a form of the Yang-Baxter equation
R_{12)(u)R_{13}(u+v)R_{23}(v) = R_{23}(v)R_{13}(u+v)R_{12}(u)
The Yang-Baxter equation is a braid relation, where the braid group B_3 which for contained in SL(2,R) in the diagram
B_3 ------- > PSL(2,Z)
∩...............|
|.................|
|.................|
v...............v
SL(2,R)--- > PSL(2,R)
Sorry for the symbols used, but the first vertical arrow is an inclusion map and the dots are used as place holders so the blog text system does not collapse it. Braid groups are constructed from knot polynomials. The knot polynomial is constructed from a gauge potential that obeys the Chern-Simons Lagrangian. The gauge potential in a Wilson line integral
W(A) = exp(∫_γA)
that defines a holonomy. The holonomy is defined by a group valued principal bundle P(G), W(A) ε P(G). This holonomy maps this path on the manifold M to a topological path or the first fundamental group π_1(M) --- > P(G). Since we are working with spacetime manifolds we set G = SL(2,C). The construction of knot polynomials is completed with the introduction of the Chern-Simons action. I will not belabor this aspect of the theory, which is fairly well known.
The Yangian is a way of representing a dual gauge theory. A scattering amplitude, in particular within the BCFW recursion relationship I discuss in my essay , is a set of external momentum lines entering a vertex. The total sum of momentum entering the vertex in the diagram is zero. This may be expressed as a polygon with edgelinks connected at different points. The differences between these points are the momentum of each line in a scattering diagram. This polygon is self dual, so each point may represent a momentum in a dual space. The two diagrams have a linear fractional relationship or SL(2,Z) transformation between them. This then describes the dual scaling that I have been mentioning.
The decomposition R^4 = R^3x{t_n}, where {t_n} means gluing these increments together, then considers each 4-manifold as a thick slice, where these foliate together. These are "thickened" 3-spaces, where we can consider the Chern-Simon's Lagrangian. There is of course another way to work this which is to decompose this as R^{3,1} = R^{2,1}x{δx_n}. In this setting the CS action makes more physical sense.
At any rate these are some thoughts and developments on my behalf in the last few days.
Cheers LC