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I will first remind you of the definition because it was rather long and so far we didn't really understand what it means.

So what is a quasi-Hamiltonian g-space or quasi-Hamiltonian g-manifold?

It's a manifold with an action of g and a map which we called mu.

There's this group-valued moment map and there is also some kind of quasi-simplistic two-form.

So it's not really a simplistic two-form but it's a two-form on a hem.

And satisfying some conditions. Probably the most important is some kind of moment map condition between...

...moment map playing between omega and mu which says what happens if you take an element of the Lie algebra and plug it into...

...so it acts on M, so you have a vector field on M and you plug it into omega.

So you can take Ix of omega.

And the rule is that you should contract x with...

...and we need to take the mean between the left and the right environment by a carton form.

So that's how it works.

So that's why we have this one-half here.

And write mu minus one d mu plus d mu mu minus one.

And then there are some further conditions which say something like that.

d of mu is equal to one-half mu star of eta where eta is some nice invariant three-form on g.

And the mu is an equivalent since we also have action of g here.

And finally there is some non-degeneracy condition which says that the kernel of omega at every point...

...does not intersect the kernel of the differential of mu.

So it's a relatively long definition.

And now we shall do some examples.

This will be the single most important example in this topic.

So let's do it.

The example somehow comes geometrically from a triangle.

So imagine you have a triangle.

I choose this kind of orientation for the edges.

And let's choose elements say h1, h2, h3.

So these hi's, they are in g.

With the condition that their product when they go around is equal to one.

And let's take the space of all these configurations.

So it means that we really take the space of all h1, h2, h3 in g cubed such that h1, h2, h3 is equal to one.

So it's a subspace of g cubed.

And in the end you see h1, h2 might be arbitrary and h3 is determined by them, it's just the inverse of the product h1, h2.

So if we wish, we can say that this is the same as g cubed.

And let us call this space M triangle.

In the end it is going to be some modular space of something.

So how should you think about these things or where is it going to come from?

So for people who just came, I said we consider now triple of elements from g that satisfy this condition that h1 times h2 times h3 is equal to one.

So where does such a thing come from?

So imagine that here on the triangle we have a g-valued connection.

So if you have a connection, then we can say we have one form in omega 1 of the triangle with values in the algebra.

So every time we choose some path, we can compute a parallel transport.

We get an element of g and suppose that our connection is flat, that the curvature vanishes.

dA plus aA is the curvature.

This vanishes, that means that if you take parallel transport along a closed curve, then we are going to get one.

And now we are just keeping only the parallel transports between the vertices.

So if we wish, we can modulate by gauge transformations, which might be completely arbitrary,

except that they should not act at these three points, meaning that those gauge transformations should be identity at these three points,

so that we don't change these three polynomials.