Grothendieck-teichmuller lie algebra pdf

The lie algebra of g is an extension of an abelian one dimensional lie algebra, by a free lie algebra. It is known that the indecomposable part is not only colie, but in fact coprelie, cf. Action of the graded grothendieckteichmueller gt group on a resolution of the operad of gerstenhaber algebras ga is defined. The conneskreimer hopf algebra is an algebra of polynomials endowed with an ad hoc coproduct, cf. Kontsevichs graph complex to the deformation complex of the sheaf of polyvector elds on a smooth algebraic variety. Is every finite group a quotient of the grothendieckteichmuller group. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Introduction the grothendieck teichmuller group grt 1 is a prounipotent group introduced by drinfeld in dr.

The grothendieckteichmuller group, the title double shuffle. On the other hand, it is known that the rational prol galois image algebraic group is embedded into the grothendieckteichmuller. On associators and the grothendieckteichmuller group i. In particular, we reprove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a notnecessarilyrational associator exists. We present a formalism within which the relationship discovered by drinfeld in dr1, dr2 between associators for quasitriangular quasihopf algebras and a variant of the grothendieck teichmuller group becomes simple and natural, leading to a simplification of drinfelds original work. If we linearly dualize as graded modules, the hopf algebra is the grossmanlarson hopf algebra, which is cocommutative and its primitive part is. This interpretation allows us to give a new description of the elements of \\widehatgt\, as well as a new proof of the drinfeldihara theorem stating.

It is proven that, for any affine supermanifold m equipped with a constant odd symplectic structure, there is a universal action up to homotopy of the grothendieck teichmuller lie algebra grt1 on the set of quantum bv structures i. On associators and the grothendieckteichmuller group i selecta mathematica, new series 4 1998 183212, june 1996, updated october 1998, arxiv. Is every finite group a quotient of the grothendieck teichmuller group. This uses a sort of profinite completion of the free braided monoidal category on one object, braid. Section 3 gives a proof of theorem 2 and its analogue in the prolgroup and pronilpotent group setting. The structure of an algebra over an e 2operad includes a product and homotopies that make this product associative and commutative in homology. Associators and the grothendieckteichmuller group 3 tij with 1.

The chord diagram operad and the rational model of operads 439 476. Let lie n stand for the free lie algebra over k generated by n letters x 1. Multiple zeta values and double shuffle lie algebra. We present a formalism within which the relationship discovered by drinfeld in dr1, dr2 between associators for quasitriangular quasihopf algebras and a variant of the grothendieckteichmuller group becomes simple and natural, leading to a simplification of drinfelds original work. Lie algebras and ados theorem princeton university. The grothendieckteichmueller lie algebra and browns dihedral moduli spaces. Physically this action is analogous to a renormalization group action. The lie algebra grt 1 is known to contain a free lie subalgebra with a generator in each odd degree g 3 bro12. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Consider a lie algebra gt generated by the variables tij, i,j.

Introduction the grothendieckteichmuller group grt 1 is a prounipotent group introduced by drinfeld in dr. A central motivating example for or special case of the study of higher algebra was. We prove these formulas on the level of lie algebras using standard techniques from the theory of vassiliev invariants and the theory of lie algebras. Derived grothendieckteichmuller group and graph complexes aftert. Grothendieckteichmuller and batalinvilkovisky core. We show that the lower central series of the latter lie algebra induces a decreasing filtration of the grothendieck teichm\uller lie algebra and we study the corresponding graded lie algebra.

Grothendieckteichmuller theory moduli spaces and multizeta. The grothendieck teichm\uller lie algebra is a lie subalgebra of a lie algebra of derivations of the free lie algebra in two generators. The graded drinfeldkohno lie algebra operads and the applications of chevalleyeilenberg cochain complexes 418 455. We show that the lower central series of the latter lie algebra induces a decreasing filtration of the grothendieckteichm\uller lie algebra and we study the corresponding graded lie algebra. The algebraic theory and its topological background. These themes lie at the forefront of current research in algebra, algebraic and differential geometry, number theory, topology and mathematical physics. Seminaire bourbaki janvier 2017 69eme annee, 20162017, n. It is an extension of the multiplicative group gm by its prounipotent radical g1. In particular, we reprove that rational associators exist and can be constructed iteratively. Little discs operads, graph complexes and grothen dieck. Note that any subspace of an abelian lie algebra is an ideal. Motives in stable homotopy theory and a conjecture of.

It is shown that the induced lie algebra action is homotopically nontrivial i. But the homology of an algebra over an e 2operad comes also equipped with a lie bracket which represents an obstruction to have a fully homotopy commutative structure at the algebra level. We also prove that, for any given homotopy involutive lie bialgebra structure on a vector space, there is an associated homotopy batalinvilkovisky algebra structure on the associated chevalleyeilenberg complex. Reminders on the drinfeldkohno lie algebra operad 446 483. Higher algebra or homotopical algebra is similarly, but in particular, the study of monoids internal to higher categories. On associators and the grothendieckteichmuller group, i. The grothendieckteichm\uller lie algebra is a lie subalgebra of a lie algebra of derivations of the free lie algebra in two generators. Quantizations are constructed by universal associators. In this article we interpret the relations defining the grothendieckteichmuller group \\widehatgt\ as cocycle relations for certain noncommutative cohomology sets, which we compute using a result due to brown, serre and scheiderer.

The operad of chord diagrams and drinfelds associators. Grothendieckteichmuller groups, multiple zeta values and certain ga lois image. We exhibit elements in kergrt ell grt and conjecture that they generate this lie algebra. We show that the action of delignedrinfeld elements of the grothendieckteichmuller lie algebra on the cohomology of. Hence these lie algebras act also on stable formality. English version of quasihopf algebras mathoverflow. Gt lie algebra appears as the tangent lie algebra to the gt group. It is known how the grothendieck teichm\uller lie algebra, or its relative, the graph complex, act on the lie algebra of polyvector fields. We compare two geometrically constructed subgroups i.

Using the explicit structure of the root category of the f. Lie algebra, the kashiwaravergne lie algebra, the double shu. Since a tangential base point corresponds to a numbered planar. Kontsevichs graph complex is isomorphic to the grothendieckteichmueller lie algebra. This action exhausts all rational automorphisms of ln up to homotopy. The implication of the pentagon equation is proved for lie series. In fact, any 1dimensional subspace of a lie algebra is an abelian subalgebra.

I was wondering where i can find a pdf of drinfelds paper quasihopf algebras, which formulated the grothendieckteichmuller group. It is known how the grothendieckteichm\uller lie algebra, or its relative, the graph complex, act on the lie algebra of polyvector fields. One tantalising area where these ideas seem to come to the surface is in the drinfeld approach to grothendieckteichmuller theory. Let g be a lie bialgebra with bracket, and cobracket call g conilpotent if for any x. The motivic galois group, the grothendieckteichmuller. Homotopy of operads and grothendieck teichmuller groups. The grothendieckteichmuller lie algebra and other animals. A cohomological interpretation of the grothendieck. Dolgushev, formality quasiisomorphism for polydifferential operators with constant coe cients.

Tower decompositions, the graded grothendieckteichmuller lie algebra and the existence of rational drinfelds associators 385 432 chapter 11. The second application is a proof that the grothendieckteichmuller. Homotopy of operads and grothendieckteichmuller groups. This cochain dgalgebra is equivalent to the dual unitary commutative dgalgebra of. Quasicoxeter algebras, dynkin diagram cohomology, and. Topology of moduli spaces of tropical curves with marked points 5 acknowledgments. In section 2 we give a proof of theorem 1 by using drinfelds gadgets. Kontsevichs graph complex and the grothendieckteichmuller. The ultimate goal of this book set is to explain that this group has a topological interpretation as a group of homotopy automorphisms associated to the operad of little 2discs, which is an. Shanghai conference on representation theory of algebras.

The grothendieckteichmueller lie algebra and browns dihedral. Kontsevichs graph complex and the grothendieckteichmueller. Grothendieckteichmuller groups, deformation and operads. The grothendieck teichmuller group g acts on the moduli space of quantum field theories. The grothendieck teichmuller group was defined by drinfeld in quantum group theory with insights coming from the grothendieck program in galois theory. T, i6 j, tij tji satisfying the relations tij,tkl 0 if i,j,k,lare all di.