Since a tangential base point corresponds to a numbered planar. Kontsevichs graph complex is isomorphic to the grothendieckteichmueller lie algebra. 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. Lie algebra case in this section we prove the lie algebra version of theorem 1 in a. Grothendieckteichmuller and batalinvilkovisky core.
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. We show that the action of delignedrinfeld elements of the grothendieckteichmuller lie algebra on the cohomology of. Is every finite group a quotient of the grothendieck teichmuller group. The second application is a proof that the grothendieckteichmuller. 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. Grothendieckteichmuller groups, multiple zeta values and certain ga lois image. Dolgushev, formality quasiisomorphism for polydifferential operators with constant coe cients. The grothendieckteichm\uller lie algebra is a lie subalgebra of a lie algebra of derivations of the free lie algebra in two generators. Willwacher 15 revealed that the generators of drinfelds grothendieckteichmuller lie algebra grt are source of at. Homotopy of operads and grothendieckteichmuller groups.
Section 3 gives a proof of theorem 2 and its analogue in the prolgroup and pronilpotent group setting. The grothendieck teichmuller group g acts on the moduli space of quantum field theories. The lie algebra grt 1 is known to contain a free lie subalgebra with a generator in each odd degree g 3 bro12. Tower decompositions, the graded grothendieckteichmuller lie algebra and the existence of rational drinfelds associators 385 432 chapter 11. Lie algebras and ados theorem princeton university. The operad of chord diagrams and drinfelds associators. We compare two geometrically constructed subgroups i. The grothendieckteichmueller lie algebra and browns dihedral moduli spaces. Lie algebra, the kashiwaravergne lie algebra, the double shu. Kontsevichs graph complex to the deformation complex of the sheaf of polyvector elds on a smooth algebraic variety. The grothendieck teichm\uller lie algebra is a lie subalgebra of a lie algebra of derivations of the free lie algebra in two generators.
We exhibit elements in kergrt ell grt and conjecture that they generate this lie algebra. In particular, we reprove that rational associators exist and can be constructed iteratively. These themes lie at the forefront of current research in algebra, algebraic and differential geometry, number theory, topology and mathematical physics. Motives in stable homotopy theory and a conjecture of. Abstract we present a formalism within which the relationship discovered by drinfeld 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. T, i6 j, tij tji satisfying the relations tij,tkl 0 if i,j,k,lare all di.
The algebraic theory and its topological background. 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. On associators and the grothendieckteichmuller group i. Using the explicit structure of the root category of the f. Reminders on the drinfeldkohno lie algebra operad 446 483. Topology of moduli spaces of tropical curves with marked points 5 acknowledgments. Introduction the grothendieckteichmuller group grt 1 is a prounipotent group introduced by drinfeld in dr. The structure of an algebra over an e 2operad includes a product and homotopies that make this product associative and commutative in homology. 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. One tantalising area where these ideas seem to come to the surface is in the drinfeld approach to grothendieckteichmuller theory. Is every finite group a quotient of the grothendieckteichmuller group. The motivic galois group, the grothendieckteichmuller. Little discs operads, graph complexes and grothen dieck.
Note that any subspace of an abelian lie algebra is an ideal. The chord diagram operad and the rational model of operads 439 476. Multiple zeta values and double shuffle lie algebra. Kontsevichs graph complex and the grothendieckteichmuller. 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.
Let g be a lie bialgebra with bracket, and cobracket call g conilpotent if for any x. Higher algebra or homotopical algebra is similarly, but in particular, the study of monoids internal to higher categories. Action of the graded grothendieckteichmueller gt group on a resolution of the operad of gerstenhaber algebras ga is defined. Seminaire bourbaki janvier 2017 69eme annee, 20162017, n. 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. Homotopy of operads and grothendieck teichmuller groups. Hence these lie algebras act also on stable formality. Introduction the grothendieck teichmuller group grt 1 is a prounipotent group introduced by drinfeld in dr. In section 2 we give a proof of theorem 1 by using drinfelds gadgets. This cochain dgalgebra is equivalent to the dual unitary commutative dgalgebra of. Let lie n stand for the free lie algebra over k generated by n letters x 1. It is known how the grothendieckteichm\uller lie algebra, or its relative, the graph complex, act on the lie algebra of polyvector fields.
The grothendieckteichmuller group was defined by drinfeld in quantum group theory with insights coming from the grothendieck program in galois theory. This action exhausts all rational automorphisms of ln up to homotopy. Quantizations are constructed by universal associators. The grothendieckteichmuller lie algebra and other animals. Grothendieckteichmuller theory moduli spaces and multizeta. Physically this action is analogous to a renormalization group action. The grothendieck teichmuller group was defined by drinfeld in quantum group theory with insights coming from the grothendieck program in galois theory. A central motivating example for or special case of the study of higher algebra was. 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.
Grothendieckteichmuller groups, deformation and operads. Quasicoxeter algebras, dynkin diagram cohomology, and. If we linearly dualize as graded modules, the hopf algebra is the grossmanlarson hopf algebra, which is cocommutative and its primitive part is. In fact, any 1dimensional subspace of a lie algebra is an abelian subalgebra. The grothendieckteichmuller group, the title double shuffle. Associators and the grothendieckteichmuller group 3 tij with 1. The grothendieckteichmueller lie algebra and browns dihedral. On the other hand, it is known that the rational prol galois image algebraic group is embedded into the grothendieckteichmuller.
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. I was wondering where i can find a pdf of drinfelds paper quasihopf algebras, which formulated the grothendieckteichmuller group. It is an extension of the multiplicative group gm by its prounipotent radical g1. This uses a sort of profinite completion of the free braided monoidal category on one object, braid. The lie algebra of g is an extension of an abelian one dimensional lie algebra, by a free lie algebra.
Kontsevichs graph complex and the grothendieckteichmueller. Shanghai conference on representation theory of algebras. It is known that the indecomposable part is not only colie, but in fact coprelie, cf. 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.
Consider a lie algebra gt generated by the variables tij, i,j. Derived grothendieckteichmuller group and graph complexes aftert. On associators and the grothendieckteichmuller group i selecta mathematica, new series 4 1998 183212, june 1996, updated october 1998, arxiv. Gt lie algebra appears as the tangent lie algebra to the gt group. It is shown that the induced lie algebra action is homotopically nontrivial i. The conneskreimer hopf algebra is an algebra of polynomials endowed with an ad hoc coproduct, cf. The implication of the pentagon equation is proved for lie series. It is known how the grothendieck teichm\uller lie algebra, or its relative, the graph complex, act on the lie algebra of polyvector fields.
198 466 1480 48 1004 990 1072 1004 1219 1631 880 1501 1311 483 267 1216 233 1475 1009 1339 641 1447 794 1011 631 1535 407 282 1239 78 441 1161 1277 568 1444 1471 1067 402 1372 393