How are motives related to anabelian geometry and galois. The applications of rational homotopy theory methods 10. The lie groups of a lie algebra mathematics for physics. Relation to deformation quantization and the cosmic galois group grothendieck predicted that the gt group is closely related to the absolute galois group. Alexander grothendieck his research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his socalled relative perspective led to revolutionary advances in many areas of pure mathematics. 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. Galoisteichmuller theory and arithmetic geometry advanced. A manifestation of the grothendieckteichmueller group in. The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship. It doesnt read as good, but it seems to be nice as a reference book. Classical teichmuller theory is a topic in complex analysis. In mathematics, the grothendieckteichmuller group gt is a group closely related to and possibly equal to the absolute galois group of the rational numbers. The book lie groups, lie algebras, and representations an elementary introduction from brian hall is a good book, as well.
Handbook of teichmuller theory irma lectures in mathematics. Gt lie algebra appears as the tangent lie algebra to the gt group. Lie theory has been particularly useful in mathematical physics since it describes the. It provides, in the classical setting of field theory, an alternative perspective to that of emil artin based on linear algebra, which became standard from about the.
I have made them public in the hope that they might be useful to others, but these are not o cial notes in any way. My understanding of these subjects is not very solid at this moment, but this is what i understand. In it, grothendieck established systematic foundations of. The graded lie algebra of the grothendieckteichmuller group. The grothendieck teichmuller group is an important and somewhat mysterious. Why havent grothendiecks books on algebraic geometry. The grothendieckteichm \ uller lie algebra is a lie subalgebra of a lie algebra of derivations of the free lie algebra in two generators. Sep 08, 2010 we show that the zeroth cohomology of m. The first aim of this book is to give an overall reference, starting from scratch, on the. 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.
In mathematics, grothendiecks galois theory is an abstract approach to the galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. Kontsevichs graph complex is isomorphic to the grothendieckteichmuller lie algebra equation. So you would need complex analysis and probably real analysis before that. 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. Sep 27, 20 for the easy part, the history, wikipedia is a good place to start. This book provides a very useful reference for known and new results about operads and rational homotopy theory and thus provides a valuable resource for researchers and graduate students interested in some of the many topics that it covers.
By grothendiecks anabelian conjectures, galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over numberfields encode all the arithmetic information of these curves. The first part of this twopart set gives a comprehensive survey on the algebraic aspects of this subject. What are topicsbooks which are essential to understanding. Anabelian geometry tries to ask how much information about a variety is contained in its etale fundamental. 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 object used to model commutative homotopy structures in topology. On associators and the grothendieckteichmuller group, i. Why is alexander grothendieck revered by mathematicians. The grothendieckteichmuller group is an important and somewhat mysterious. The grothendieckteichmueller lie algebra and browns dihedral. The grothendieckteichmueller lie algebra and browns dihedral moduli spaces. On associators and the grothendieck teichmuller group i selecta mathematica, new series 4 1998 183212, june 1996, updated october 1998, arxiv.
Brookes part iii course on lie algebras, given at cam bridge university in michaelmas term, 2012. Alexander grothendieck introduced many concepts into algebraic geometry. Kontsevichs graph complex and the grothendieckteichmueller. What perhaps less wellknown is that this result can be interpreted purely in terms of the fundamental group. The goal of this paper is to develop an arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic. Derived grothendieckteichmuller group and graph complexes aftert. In recoltes et semailles, grothendieck remarks that the theory of motives is related to anabelian geometry and galois teichmuller theory. The grothendieckteichmuller group was defined by drinfeld in quantum group theory with insights coming from the grothendieck program in galois theory. Some time ago the story used to be told was that grothendieck put it on a rigorous footing after the italian school starting in the 19th century castelnuovo gradually neglected rigor enriques, and ended up with results which no one. If the number of degrees of freedom is finite, so is the set of possible distinct measurement devices one can construct.
Introduction it is wellknown that teichmiiller space is a ball of dimension 69 6. From the 1980s, grothendiecks esquisse dun programme triggered tremendous developments in number theory and arithmetic geometry, extending from the studies of anabelian geometry and related galois representations to those of polylogarithms and multiple zeta values, motives, rational points on arithmetic varieties, and effectiveness. The grothendieckteichmuller group was defined by drinfeld in quantum. We compare two geometrically constructed subgroups i. The ams bookstore is open, but rapid changes related to the spread of covid19 may cause delays in delivery services for print products. Laubinger on lie algebras for frolicher groups the n. In particular, we reprove that rational associators exist and can be constructed iteratively. Jun 04, 2015 the goal of this paper is to develop an arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same. The latter group studies the fine details of the terrain. Grothendieck is arguably the most brilliant mathematician of the 20th century, with his influence felt the most in algebraic geometry, which he transformed.
Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase. This was established for the class of leavitt path algebras associated to polycephaly graphs 3. The operadic viewpoint is thoroughly explained in fresses book 22, see also merkulovs. We will see why grothendieck wrote to serre on february 18, 1955. Sur quelques points dalgebre homologique, ii grothendieck, alexander, tohoku mathematical journal, 1957. This lie algebra is graded by positive integers and, 8 odd n 3 9a nonzero element. Pentagon and hexagon equations annals of mathematics. By grothendieck s anabelian conjectures, galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over numberfields encode all the arithmetic information of these curves. Furthermore, the lie algebra is introduced not as an abstract linear space with a bracket but as the set of.
Lie groups evolve out of the identity 1 and the tangent vectors to oneparameter subgroups generate the lie algebra. Nielsen book data the ultimate goal of this book is to explain that the grothendieckteichmuller group, as defined by drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2disc operad. In mathematics, grothendieck s galois theory is an abstract approach to the galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. For the easy part, the history, wikipedia is a good place to start. Homotopy of operads and grothendieckteichmuller groups. This uses a sort of profinite completion of the free braided monoidal category on one object, braid. Homotopy of operads and grothendieckteichmuller groups in. This result has applications to deformation quantization and duflo theory. Jun 30, 2017 the ultimate goal of the second part of the book is to explain that the grothendieckteichmuller group, as defined by drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2disc operad. One tantalising area where these ideas seem to come to the surface is in the drinfeld approach to grothendieckteichmuller theory. The basic example of a lie algebra arises from using the commutator in an associative algebra, so x. The goal of this paper is to develop an arithmetic teichmuller theory. On associators and the grothendieckteichmuller group i selecta mathematica, new series 4 1998 183212, june 1996, updated october 1998, arxiv.
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. 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. It is conjectured that several graded lie algebras coming up in different fields of mathematics coincide. The grothendieckteichmuller lie algebra is isomorphic to the 0th cohomology of kontsevichs graph complex willwacher 10. The goal of this paper is to develop an arithmetic teichmuller theory, by which we mean, introducing arithmetic objects. If a m nk, then the space of n nmatrices has the structure of a lie algebra with lie bracket x. Kontsevichs graph complex and the grothendieckteichmuller. Drinfeld formally introduced two grothendieck teichmuller groups, the former one related to the absolute galois group and the latter one related to the deformation theory of a certain algebraic structure braided quasihopf algebra.
Grothendieck rings of basic classical lie superalgebras 665 variation of the algebra of invariant polynomials investigated for lie superalgebras in 2, 14, 23, and 24. Drinfeld formally introduced two grothendieckteichmuller groups, the former one related to the absolute galois group and the latter one related to the deformation theory of a certain algebraic structure braided quasihopf algebra. In recoltes et semailles, grothendieck remarks that the theory of motives is related to anabelian geometry and galoisteichmuller theory. Why havent grothendiecks books on algebraic geometry been. What perhaps less wellknown is that this result can be interpreted purely in terms of. Grothendieck teichmuller and batalinvilkovisky sergei merkulov and thomas willwacher abstract. The terms that continue the series are all expressed in terms of lie commutators, and as lie brackets hold for the exponential maps of any lie algebra. 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.
The graded grothendieck group as a classification tool for algebras, current status it is conjectured 3 that the graded grothendieck group is a complete invariant for leavitt path algebras. Complete lie algebras and models of classifying spaces. Alexander grothendieck was is a genius of the first order, and a truly amazing spirit. It is known how the grothendieck teichm\uller lie algebra, or its relative, the graph complex, act on the lie algebra of polyvector fields. Depending on the approach you take to teichmuller theory, some knowledge of manifolds and differential geometry would also be helpful. The structure of a lie group is implicit in its algebra, and the structure of the lie algebra is expressed by root systems and root data. It is proven that, for any a ne supermanifold m equipped with a constant odd symplectic structure, there is a universal action up to homotopy of the grothendieck teichmuller lie algebra grt 1 on. In more abstract terms, there is a bijection between the set of associators and a certain morphism of operads. We also compute the homotopy derivations of the gerstenhaber operad. The ultimate objective of this book is to prove that the grothendieckteichmuller group is the group of. Freeman dyson once categorized mathematicians as being of roughly two types.
Why did algebraic geometry need alexander grothendieck. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Let x be a smooth algebraic variety over an algebraically closed. 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. Grothendieck rings of basic classical lie superalgebras. Grothendieckteichmuller groups, deformation and operads.
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