the necklace Lie bialgebra

Today Travis Schedler posted a nice paper on the arXiv “A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver”. I heard the first time about necklace Lie algebras from Jacques Alev who had heard a talk by Kirillov who constructed an infinite dimensional Lie algebra on the monomials in two non-commuting variables X and Y (upto cyclic permutation of the word, whence ‘necklace’). Later I learned that this Lie algebra was defined by Maxim Kontsevich for the free algebra in an even number of variables in his “Formal (non)commutative symplectic geometry” paper (published in the Gelfand seminar proceedings 1993). Later I extended this construction together with Raf Bocklandt in “Necklace Lie algebras and non-commutative symplectic geometry” (see also Victor Ginzburg’s paper “Non-commutative symplectic geometry, quiver varieties and operads”. Here, the necklace Lie algebra appears from (relative) non-commutative differential forms on a symmetric quiver and its main purpose is to define invariant symplectic flows on quotient varieties of representations of the quiver.
Travis Schedler extends this construction in two important ways. First, he shows that the Lie-algebra is really a Lie-bialgebra hence there is some sort of group-like object acting on all the representation varieties. Even more impoprtant, he is able to define a quantization of this structure defining a Hopf algebra. In this quantization, necklaces play a role similar to that of (projected) flat links in the plane whereas their quantization (necklaces with a height) are similar to genuine links in 3-space.
Sadly, at the moment there is no known natural representations for this Hopf algebra playing a similar role to the quotient varieties of quiver-varieties in the case of the necklace Lie bialgebra.

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