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.

# the necklace Lie bialgebra

Published in featured

## Comments