Abel's equation for the dilog - MathOverflow

Abel's equation for the dilog

2009-10-25T00:53:03Z

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms) plays a role in web geometry as it is one of the abelian relations of the first example of exceptional web (Bol's 5-web) to appear in the literature.

I have heard it is important in other domains (cohomology of SL(3,C), algebraic K-theory, motives ). I would like to learn more about it.

I am asking for:

Insights on why Abel's identity is relevant in this or that field;
References where it plays a role.

Edit. I have just learned from this blog about Bridgeman's orthospectrum identity. Those interest in the question above might want to take a look at it.

Answer by Rob Harron for Abel's equation for the dilog

2009-10-25T02:59:14Z

For the relation to motives (and K-theory), I'd suggest the first several article of Motives, volume 2 (the proceedings of the Seattle conference link text). I don't really know this stuff, but it is apparently believed that the polylogarithms are related to the "higher regulators" from K-theory to Deligne cohomology. These regulators are supposed to help explain the values of L-functions of motives at integers. Apparently the usual logarithm occurs in the first chern class of a variety (and the regulators are thought of as generalizations of chern class, or something). Good luck.

Answer by Lavender Honey for Abel's equation for the dilog

2009-10-25T04:23:25Z

One basic answer is given by hyperbolic geometry.

Ideal tetrahedra in hyperbolic 3-space H^3 are equivalent (under the action of the automorphism group PGL_2(C)) to tetrahedra with vertices {0,1,oo,z}, and their volume is given by D(z), where D(z) is the Bloch-Wigner dilogarithm, which is a slightly modified version of the dilogarithm. This amounts to writing down the hyperbolic metric and evaluating an integral, which turns out to be (very close to) Li_2(z) (although it is real valued for complex z).

The tetrahedron {0,1,oo,z} is equivalent under PSL_(2)(C) to {0,1,oo,1/(1-z)} and {0,1,oo,1-1/z}, and so we get formulae:

D(z) = D(1/(1-z)) = D(1 - 1/z).

The tetrahedron {0,1,oo,z} is also equivalent to {0,1,oo,1/z}, except with an odd permutation of the vertices, and thus:D(z) = - D(1/z).

Finally, choose a random point y in the boundary P^1(C) of H^3. If we take the tetrahedron {0,1,oo,y}, we can break it off into {0,1,oo,x} and three other tetrahedra (just like in Euclidean space). Transforming the coordinates of the other three tetrahedra into the standard form gives the 5-term relation:

D(x) - D(y) + D(y/x) - D((1-1/x)/(1-1/y)) + D((1-x)/(1-y)) = 0,which gives a proof of Abel's equation.

Let's think some more about a closed hyperbolic 3-manifold M. By definition, M = H^3/Gamma for a lattice Gamma in PSL_2(C). Since H^3 is contractible, M is a K(Pi,1) space, and so there is a canonical isomorphism H_*(M,Z) = H_*(Gamma,Z), comparing simplicial homology with the group homology of Gamma. Now M has a fundamental class [M] in H_3(M,Z), which gives an element in H_3(Gamma,Z) and hence also a class in H_3(PSL_2(C),Z). On the other hand, [M] can be decomposed ("triangulated") into ideal tetrehedra with parameters z_i. The set of parameters [z_i] is not unique, however, the only real "move" is the subdivision of tetrahedra, and so associated to M we get an element of the group generated by [z_i] for z_i in P^1(C) and with relation s exactly of the form satisfied by D above. This is essentially the definition of the Bloch group. D is a function this group, and this decomposition gives a map from H_3(PSL_2(C),Z) to the Bloch group.

Note that it is not obvious that the z_i can be taken inside some field F, this is a consequence of Mostow Rigidity. It turns out that if we take the Bloch group B(F) generated by elements of F, this is, by work of Suslin, essentially equal to K_3(F).

To summarize, the connection between the identity, the cohomology of PSL_2(C), and the Bloch group is well understood, see some papers by Walter Neumann. For the connection between the Bloch group B(F) and K_3(F), see papers of Suslin. The connection with motives is more speculative, but here you should look at some papers of Goncharov.

(There are some generalizations/connections to higher regulators for K-groups, but this is a very nice example to understand, being both somewhat accessible yet still very interesting.)

If I could, I would add a flag to this post "hyperbolic geometry".

Answer by Andreas Holmstrom for Abel's equation for the dilog

2009-10-25T04:32:05Z

There is a little book by Bloch, called Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. It was published quite recently but is based on a famous lecture series from the late 70s or so. He treats the dilog specifically, rather than the more general polylog framework referred to by Rob. See in particular chapter 6.

Answer by Wadim Zudilin for Abel's equation for the dilog

2010-03-27T13:07:11Z

There is a remarkable article, "The remarkable dilogarithm," J. Math. Phys. Sci. 22 (1988), 131--145, by Don Zagier, which was recently reprinted and updated as "The dilogarithm function" (63 pages!) in one of the collections by Springer Verlag.