Lie algebroids, holonomy and characteristic classes rui loja fernandes dep. Dms, kohno, gawedzkiwzw a ne lie kacmoody algebras. This can be studied with the tools of riemannian geometry, leading to consequences in the theory of holonomy. Our next objective is to find a presentation for the holonomy lie algebra h g. For any vector space v we have slv, the special linear lie algebra of v, consisting of the operators on v of trace 0.
For example, a graded lie algebra is a lie algebra with a graded vector space structure. Embedding of the virasoro algebra sugawara construction. This description of the holonomy algebra proves to be an important tool for the classi. Ais the vertex set of a, mv denotes the multiplicity or degree of the vertex v, and tvd q i2v ti. Furthermore, some infinitedimensional lie algebras are not the lie algebra of any group. Let x belong to the lie algebra g of g, v be a principal vector and. In view of the first relation in the drinfeldkohno lie algebra, it suffices to. In mathematics, lie algebra cohomology is a cohomology theory for lie algebras. The monodromy of rin x is the quotient group holxhol0x. Simons collaboration on special holonomy in geometry, analysis and physics second annual meeting on simons foundation. Chen 6, and has been further studied in a number of papers, including 24. Let g be a complex semi simple lie algebra and i be an orthonormal basis of g with respect to the. Holonomies for connections with values in l 1algebras camilo arias abad and florian sch atzy april 28, 2014 abstract given a at connection on a manifold m with values in a ltered l 1 algebra g, we construct a morphism hol1. Cup products, lower central series, and holonomy lie algebras.
Quasicoxeter algebras, dynkin diagram cohomology, and. Quantum and homological representations of braid groups. It is known that the holonomy group of a flat solvmanifold is abelian. Holonomy lie algebras and the lcs formula for subarrangements of. Lie algebra homology and cohomology shenning tung november 26, 20 abstract in this project we give an application of derived functor. Our goal is to make an attempt to nd the right notion of the holonomy algebra. We construct a representation of the homotopy 2groupoid of a manifold by means of k. Kohno, t on the holonomy lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. Systems, holonomy lie algebras and logarithmic vector fields 5 theorem 1. Monodromy representations of conformal field theory. On the holonomy lie algebra and the nilpotent completion.
Quantum representations of braid groups and holonomy lie algebras kohno, toshitake, 2017. The second chapter brings the structure of the semisimple lie algebras. If m is simplyconnected, then the holonomy group agrees with the restricted holonomy group, and hence. As an application, we determine the chen lie algebras of several classes of geomet. Higher holonomy maps for hyperplane arrangements springerlink. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain c.
Cup products, lower central series, and holonomy lie. The boundary manifold of a complex line arrangement. This extends kohno s result 10 for braid arrangements, and provides the first instance of an lcs formula for arrangements which are not decomposable or hypersolvable. The boundary manifold of a complex line arrangement 107 where v. This extends kohno s result 10 for braid arrangements, and provides an instance of an lcs formula for arrangements which are not decomposable or hypersolvable. Koszul dgalgebras arising from configuration spaces. It is shown that the description of certain class of representations of the holonomy lie algebra g. Bu 1g, generalizing the holonomies of at connections with values in lie algebras.
Now we illustrate what the lie bracket has to do with second order effects and group commutators. A note on the lie algebras of algebraic groups kanno, tsuneo, tohoku mathematical journal, 1958. The cohomology ring of the real locus of the moduli space of stable. For example, a graded lie algebra is a lie algebra. On the holonomy lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. A geometric proof of the berger holonomy theorem by carlos olmos. By using the idea of this 2holonomy map, we describe a method to obtain a representation of the category of braid. Suciu, cohomology rings and nilpotent quotients of real and. Section 4 contains construction of further tangent lie algebras to the holonomy group consisting of of tangent vector elds on the indicatrix, namely the in nitesimal holonomy algebra and the holonomy algebra of a finsler manifold. There is a theorem ados theorem2 that any lie algebra g is isomorphic to a lie subalgebra of gln, so if you understand everything about gln, youre in pretty good shape. In this paper, we use the holonomy lie algebra to obtain a formula for when a is a subarrangement of a n. Oda, the lower central series of the pure braid group of an algebraic curve, advances in pure math. Holonomy groups of compact flat solvmanifolds springerlink. Introduction to representation theory of braid groups.
The holonomy group of the obata connection is contained in g l n, h. Citeseerx holonomy lie algebras and the lcs formula. In this article we study the holonomy groups of flat solvmanifolds. A z l is the lie algebra of a connected group kzl of linear transformations of tpz. Kr ahmer u glasgow knizhnikzamolodchikov and drinfeld kohno warsaw 8. A lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. Lie algebras alexei skorobogatov march 20, 2007 introduction for this course you need a very good understanding of linear algebra. Holonomy of braids and its 2category extension toshitake kohno the university of tokyo july, 2015 monte verit a.
The following result by kohno 6 highly increases the interest to study k has characteristic zero. The special linear lie algebra sln,r consists of all n. M for every left gmodule m, by regarding m as a ug. Lie algebroids, holonomy and characteristic classes. In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudoriemannian manifold whose group of symmetries contains an inversion symmetry about every point. Also, lis not nilpotent, because otherwise zl would be a nontrivial. Kohno in, building on work of chen, and has been further studied in a number of papers, including. As holxre is a lie subgroup of glex, it has a lie algebra hol xre, which is a lie subalgebra of ende. Schenck, the holonomy lie algebra of ageometric lattice, preprint, 2006. Vsystems, holonomy lie algebras and logarithmic vector fields. If x is the complement of a hypersurface in pn, then kohno showed that the nilpotent completion of the fundamental group is isomorphic to the nilpotent completion of the holonomy lie algebra of x.
Hyperplane arrangements, local system homology and iterated. Lower central series, free resolutions, and homotopy lie algebras of arrangements. On the way to his classification of riemannian holonomy groups, berger developed two criteria that must be satisfied by the lie algebra of the holonomy group of a torsionfree affine connection which is not locally symmetric. Twisted alexander polynomials inform on invariants such as the alexander and thurston.
S1 the virasoro lie algebra acts on h toshitake kohno monodromy representations. In this paper we show that the holonomy of the obata connection on s u 3 coincides with g l 2, h. Orliksolomon algebras in algebra and topology iopscience. Kohno, on the holonomy lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces, nagoya math. They showed similarly that 2transport functors from the thin fundamental 2groupoid correspond to pairs of di 1erential forms a 2 m. The degree zero part of the graded lie algebra contains the associated kacmoody algebra. Bu 1g, generalizing the holonomies of at connections with values in lie. Knizhnikzamolodchikov connection on the con guration space of points kz. We generalize basic results relating the associated graded lie algebra and the holonomy lie algebra from finitely presented, commutatorrelators groups to arbitrary finitely presented groups.
A proper representation theory is then provided using the gelfand spectral theory. Suciu, lower central series and free resolutions of hyperplane arrangements, arxiv. Chapter 12, classification of connected coxeter graphs. A corollory of these general results is a precise formulation of the loop transform proposed by rovelli and smolin. Lower central series, free resolutions, and homotopy lie. The basic example of a lie algebra arises from using the commutator in an associative algebra, so x. If x is the complement of a hypersurface in 8n, then kohno showed in 11 that the nilpotent completion of. We describe a relationship between linear representations of the braid groups due to r. In the process, we give an explicit formula for the cupproduct in the cohomology of a finite 2complex, and an algorithm for computing the corresponding holonomy lie algebra, using a magnus. In this paper, we use the holonomy lie algebra to obtain a formula for. Kac v 1993 infinitedimensional lie algebras mir, moscow russian transl. L would be a nontrivial ideal or lwould be abelian.
Endv to be the lie algebra of the holonomy group of a torsion free connection, and used it to classify all irreducible nonsymmetric holonomy algebras of riemannian metrics, i. This extends kohno s result 12 for braid arrangements, and provides the first instance of an lcs formula for arrangements that are not decomposable or hypersolvable. Solutions to problems in lie algebras in particle physics by howard georgi 3 chapter 8 solutions 8. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations for example, matrix. The quantum holonomy algebra is mnstructed, and, in the case of real mnnections. The drinfeld kohno theorem for any complex lie algebra g and any. We construct a representation of the homotopy 2groupoid of a man. Simple 3dimensional lie algebras suppose lis a simple 3dimensional lie algebra over k, where kis algebraically closed. This extends kohnos result 10 for braid arrangements, and provides the first instance of an lcs formula for arrangements which are not decomposable or hypersolvable.
The simple roots are the positive roots that cannot be written as the sum of other positive. Bigelow and the holonomy representations of the kz connection. The drinfeld kohno theorem for any complex lie algebra g. Simons collaboration on special holonomy in geometry.
Vsystems, holonomy lie algebras and logarithmicvector elds. A nilpotent lie algebra with nilpotent automorphism group dyer, joan l. Therefore, it induces a monodromy representation of. There is a wellknown construction of hypercomplex structures on lie groups due to joyce. Su3 holonomy metrics from nilpotent lie groups conti, diego, asian journal of mathematics, 2014. Infinitesimal motions of a riemannian manifold 529 in 4 we consider f0 the lie algebra of skewsymmetric endomorphisms of v0 generated by all ax for xg8 0ne knows that in general fi0cg0 where kso is the normalizer of 60 in the lie algebra, a0, of all skewsymmetric endomorphisms of v0. Higher holonomy of formal homology connections and braid cobordisms toshitake kohno abstract. A and the corresponding holonomy lie algebra g with. Pdf the set of 2flats of size at least three of a simple matroid on a finite set e is the basis for defining the holonomy lie algebra. The number of generators is the dimension of the lie algebra. The corresponding arrangement is called the braid arrangement, since its fundamental group is known to be the pure braid group. Pdf the lower central series of a fibertype arrangement. This lie algebra is a quite fundamental object, that crops up at many places, and thus its representations are interesting in themselves. The results stated without proof have a geometric proof eventually, with simple.
Holonomies for connections with values in l algebras. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. Let g be a complex semisimple lie algebra and i be an orthonormal basis of g with respect to the. Riemannian holonomy groups and calibrated geometry dominic joyce notes for graduate lecture course, mt 2006.
Dec 09, 2015 pdf the set of 2flats of size at least three of a simple matroid on a finite set e is the basis for defining the holonomy lie algebra. This lie algebra coincides with the lie algebra of the unipotent completion of the fundamental. T on the holonomy lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. A simple lie algebra is a lie algebra that contains no proper ideal no proper. Holonomy of braids and its 2category extension toshitake kohno the university of tokyo july, 2015 monte verit a 128. If a m nk, then the space of n nmatrices has the structure of a lie algebra with lie bracket x. Holonomy of the obata connection on su3 arxiv vanity. Another lie algebra associated to a group g is the holonomy lie algebra, hg, which was introduced in work of chen 8, kohno 19, and marklpapadima 24. The holonomy group of a connection is closely related to its curvature. Veselov loughborough university igor krichever 60, new york, may 6, 2011.
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