Pdf a free differential lie algebra for the interval researchgate. A dg morphism between dgalgebras is a graded algebra homomorphism which respects the differential d. Suppose that g is the lie algebra of a lie group g. A strict lie 2 algebra is a category l internal to liealg.
If a m nk, then the space of n nmatrices has the structure of a lie algebra with lie bracket x. Leftinvariant modules over a group given a vector bundle f gover gwhere ghas a representation on f, a left g. We now come to the concept of a dg coalgebra, which can be viewed as the dual to the notion of a dg algebra, in a sense which will be made precise later. In these lectures we will start from the beginning the theory of lie algebras and their representations. Another way to calculate the dg lie algebra g can be described as follows. This lie algebra is a quite fundamental object, that crops up at many places, and thus its representations are interesting in themselves.
Remarks on deformation quantization of vertex poisson algebras. By the poincarebirkhoffwitt theorem it is the same size as the symmetric algebra of the free lie algebra meaning that if both sides are graded by giving elements of x degree 1 then they are isomorphic as graded vector. Related mathematical concepts include lie groups and differentiable manifolds. A differential graded augmented algebra also called a dga algebra, an augmented dg algebra or simply a dga is a dg algebra equipped with a dg morphism to the ground ring the terminology is due to henri cartan.
In particular we give a new construction of periodic and negative cyclic homology, and we construct a natural extension of the necklace lie algebra 10. Contents preface pagexi 1 introduction 1 2 liegroups. Thus l consists of a lie algebra of objects l 0, and a lie algebra of morphisms l 1 such that all structure maps are lie algebra homomorphisms. An algebra is an algebraic structure in its own right and is not to be confused with the subject called algebra. Since all of the modules at each level of b are free over our base ring ct, we say that b is a quasifree. Journal of algebra 282 2004 490512 493 from now on and till section 5. The free lie algebra on x is the lie subalgebra in tw x generated by x. The basic example of a lie algebra arises from using the commutator in an associative algebra, so x. In this paper, we show that the derived picard group of a homologically smooth koszul connected cochain dg algebra is isomorphic to the opposite group of the derived picard group of its finite. We also show how atiyah classes of lie pairs and dg lie algebroids give rise to atiyah classes of ddgalgebras. Finally, there is a free dg lie algebra functor define as follows. The product of dg lie algebras mathematics stack exchange. Notes for math 261a lie groups and lie algebras march 28, 2007 contents contents 1 how these notes came to be 4 dependence of results and other information 5 lecture 1 6 lecture 2 9 tangent lie algebras to lie groups 9 lecture 3 12 lecture 4 15 lecture 5 19 simply connected lie groups 19 lecture 6 hopf algebras 24 the universal enveloping. Lie 2algebras were introduced by baez and crans in hda6.
If k k is a field and \mathcalc svect is the category of super vector spaces over k k, then a lie algebra object is a super lie algebra. For v v a vector space, the general linear lie algebra or endomorphism lie algebra. Modular lie algebras pdf 74p this note covers the following topics. These notes are an introduction to lie algebras, algebraic groups, and lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Similarly, the free differential graded lie algebra over k on. Dg polynomial algebras, trivial dg free algebras and several nontrivial dg downup algebras and dg free algebras. A lie algebra is a vector space, l, above, the linear combinations of the ts together with a bilinear operation from l. In other words, one can construct canonically a dg associative algebra da, together with a quasiisomorphism of complexes da. Dg algebra, koszul, homologically smooth, derived picard group. Let p be an associative free dg algebra resolution of a. A differential graded lie algebra, or dgla for short, is a dg kmodule with a graded lie bracket. Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations.
Theorem for such an operad and to give an explicit small quasifree resolution for it. Operads utrecht university repository universiteit utrecht. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. The term lie algebra after sophus lie was introduced by hermann weyl in the 1930s. Here, the dg algebra structure on ca is given by the di. Take the tensor algebra tensv and look at the dg subspace v v 1. One can thus give an axiomatic characterization of the category of representations of a hopf algebra, and this is one of the most fruitful ways to deal with quantum groups. On the free lie algebra lv, the bracket length filtration comes from a gradation.
An introduction to matrix groups and their applications andrew baker 1472000 department of mathematics, university of glasgow, glasgow g12 8qw, scotland. Deformations of path algebras of quivers with relations. We are ready to connect dg lie algebras to the version of deformation theory that we met in the. The universal enveloping algebra of a free lie algebra on a set x is the free associative algebra generated by x. Free algebras, universal enveloping algebras, p th powers, uniqueness of restricted structures, existence of restricted structures, schemes, differential geometry of schemes, generalised witt algebra, filtrations, witt algebras are generalised witt algebra, differentials on a scheme, lie algebras of cartan type, root. V \mathfrakglv of v v is the lie algebra whose elements are linear endomorphisms v v v \to v and whose lie bracket is given by the commutator of endomorphisms. P of graded algebra derivations of p form a dg lie algebra. An introduction to matrix groups and their applications. Chapter 12, classification of connected coxeter graphs. Pdf derived picard groups of homologically smooth koszul. For any vector space v we have slv, the special linear lie algebra of v, consisting of the operators on v of trace 0.
Pdf derived picard groups of homologically smooth koszul dg. Mar 14, 2014 the gauge action, dg lie algebra and identities for bernoulli numbers. Atiyah classes and dglie algebroids for matched pairs. Multiplicityone representations of divergencefree lie algebras. Dg modules for dg lie algebras the enveloping algebra of a lie algebra recognizes quasiisomorphisms of dg lie algebras in the following way. Lecture 5 lie algebra cohomology ii january 28, 20 1 motivation. The second chapter brings the structure of the semisimple lie algebras. Uh is a quasiisomorphism of associative dg algebras. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. Pdf the gauge action, dg lie algebra and identities for.
In the language of deformation theory via dg lie algebras see 6, the gerstenhaber bracket is a dg lie bracket on the hochschild complex and deformations of the associative multiplication on aare given by maurercartan elements of this dg lie algebra. Introduction vertex poisson algebras deformation problem operads in 15 minutes chiral algebras chiral dg lie algebra thus a deformation quantization of a poisson algebra is the problem to ndclassify solutions. In this video, we define the notion of a lie algebra. Pdf using geometric methods we constructively show the existence of a unique completed free differential graded lie algebra modelled on.
Free lie algebra books download ebooks online textbooks. Baez and crans go further and consider the notion of a semistrict lie 2. Where xcan be canonically embedded into w x via the map. When the lie pair comes from a matched pair of lie algebroids, we show that the inclusion. The functor free l is left adjoint to the forgetful functor oblv l. A differential graded lie algebra, or dglie algebra for short. In older texts, the name infinitesimal group is used. Lie algebras alexei skorobogatov march 20, 2007 introduction for this course you need a very good understanding of linear algebra.
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