In this article character groups of hopf algebras are studied from the perspective of infinite dimensional lie theory. A pro lie group is a projective limit of a family of finite dimensional lie groups. Infinitedimensional lie groups without completeness. Projective unitary representations of infinitedimensional lie groups janssens, bas and neeb, karlhermann, kyoto journal of mathematics, 2019. The material covered ranges from basic definitions of lie groups to the classification of finitedimensional representations of semisimple lie algebras. Introduction to finite and infinite dimensional lie super. Developments and trends in infinitedimensional lie theory. A group object in infinitedimensional smooth manifolds is an infinitedimensional lie group if the underlying type of manifolds is banach manifold, then one speaks of banach lie groups, etc examples. The purpose of this paper is to survey the theory of regular frechetlie groups developed in 110. In this case the relation between the lie algebra and the lie group becomes rather subtle, and several results about finitedimensional lie groups no longer hold.
Symmetries and infinite dimensional lie algebras 69 2 groups i assume that most of the readers know the denition of a group. We give a formulation in the framework of vertex algebras. Algebras and groups 1 introduction 3 on the structure of hopf algebras preprint, 1959 with j. Integrating unitary representations of infinitedimensional. In this paper we propose a new treatment about infinite dimensional manifolds, using the language of categories and functors. We call a central zextension of a group g weakly universal for an abelian group a if the correspondence assigning to a homomorphism za the corresponding aextension yields a bijection of extension classes. In 14, 15 omori, maeda and yoshioka gave conditions under which a smooth lie group modelled on frechet spaces admits exponential. Notice that we are not even considering the more di cult topological approach of working with bg. Section 3 concerns the homology theory of infinite dimensional manifolds. Pdf ebook infinite dimensional groups and algebras in quantum physics lecture notes in physics monographs, by johnny t. In this paper we show that a prolie group g is a lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if g is locally contractible. For a graded and connected hopf algebra we construct an infinitedimensional.
Terminology and notation can mostly be found in remarks and definitions. The main problem discussed in this paper is the existence of central lie group extensions of a connected lie group g which is weakly universal for all abelian lie groups whose. Remarks on actions on compacta by some infinite dimensional groups. While discussing all classes of finite and infinite dimensional lie algebras and lie superalgebras in terms of their different classes of root systems, the book focuses on kacmoody algebras. Character groups of hopf algebras as infinitedimensional lie. Cornelia vizman, some remarks on the quantomorphism group. Walter describes the construction of lie groups c l w u, h modeled on the space c l w u, lh, where u is an open subset of a finitedimensional space x and w is a set of weights on. Projective unitary representations of infinite dimensional lie groups janssens, bas and neeb, karlhermann, kyoto journal of mathematics, 2019. Infinite dimensional manifolds from a new point of view.
Lectures on infinite dimensional lie groups mathipm. Notes on regularity properties of infinitedimensional lie groups. Induced representations and frobenius reciprocity 563. Remarks on infinite dimensional lie groups, relativity, groups and topology. The theory of infinitedimensional lie groups and its. Pdf infinitedimensional lie groups and algebras in. Control systems on lie groups heat kernels on lie groups lie groups lie group structures on quotient groups and universal complexifications for infinitedimensional lie groups chapter 0. Schmid, the differentiable structure of three remarkable diffeomorphism groups, math. Lie bracket making vectm an infinitedimensional lie algebra. In this note we show that a prolie group g is a lie group. Chernweil theory for certain infinite dimensional lie groups 3 expressions.
Lie algebra cohomology for infinite dimensional lie. Remarks on infinitedimensional lie groups, relativity. The purpose of this paper is to survey the theory of regular frechet lie groups developed in 110. Chernweil theory for certain infinitedimensional lie groups 3 expressions. Lie group structures on quotient groups and universal complexifications for infinitedimensional lie groups. Lie algebras by shlomo sternberg the campbell baker hausdorff formula, sl2 and its representations, classical simple algebras, engelliecartanweyl, conjugacy of cartan subalgebras, simple finite dimensional algebras, cyclic highest weight modules, serres theorem, and more. Other readers will always be interested in your opinion of the books youve read. Walter describes the construction of lie groups c l w u, h modeled on the space c l w u, lh, where u is an open subset of a finite dimensional space x and w is a set of weights on. A prolie group is a projective limit of a family of finitedimensional lie groups.
Such groups appear and are useful in symplectic geometry and the theory of primitive infinite groups of lie and cartan 11. Infinite dimensional lie group article about infinite. Remarks on infinitedimensional lie groups, relativity, groups and topology. Let g be a lie group which is the union of an ascending sequence g1. The articles in part c develop new, promising methods based on heat kernels, multiplicity freeness, banachliepoisson spaces, and infinitedimensional generalizations of reductive lie groups. Remarks on infinite dimensional lie groups inspirehep. This includes diffeomorphism groups and their natural subgroups like volumepreserving and symplectic transformations, as well as gauge groups and loop groups. Advanced study 1985 51 pages in les houches 1983, proceedings, relativity, groups and.
Neeb february 2, 2008 abstract aprolie group isa projective limit ofa familyof. The analytic representation theory of infinitedimensional lie groups is still very much underdeveloped. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. The relationship between infinite dimensional lie algebras and lie groups, which is. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. Many infinitedimensional lie groups of interest can be expressed as a union of an ascending sequence of finite or infinitedimensional lie groups. Home remarks on infinite dimensional lie groups citations information. We study the question when g lim gn in the category of lie groups, topological groups, smooth manifolds, resp. This book constitutes the proceedings of the 2000 howard conference on infinite dimensional lie groups in geometry and representation theory. Ergebnissedermathematik volume51 undihrergrenzgebiete. Moore 37 remarks on infinitedimensional lie groups 1986 91 the representation rings of some classical groups 1963 143 growth of finitely generated solvable groups 1968 155.
Remarks on actions on compacta by some infinitedimensional groups. Milnors exposition 8 describes the basic ideas and references up to 1984. Regular infinite dimensional lie groups heldermannverlag. One reason for study ing such groups is that they are the simplest examples of infinite dimensional lie groups. The inverse function theorem of nash and moser, bull. Applications include fluid dynamics, maxwells equations, and plasma physics. This picture is different for infinitedimensional groups, and how bad it becomes. Lectures on lie groups and geometry imperial college london. Infinitedimensional lie algebras, theta functions and. Infinite dimensional lie theory from the point of view of. Among their topics are spherical varieties and perspectives in representation theory, on toric degenerations of flag varieties, derived categories of quasihereditary algebras and their derived composition series, bounded and semibounded representations of infinite dimensional lie groups, and geometric invariant theory for principal three dimensional subgroups acting on flag varieties.
Donaldson march 25, 2011 abstract these are the notes of the course given in autumn 2007 and spring 2011. Ottesen that we provide will stimulate you to be smarter. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. We construct and study various dual pairs between finite dimensional classical lie groups and infinite dimensional lie algebras in some fock representations.
Remarks on infinite dimensional lie groups 1983 citeseerx. The literature is not entirely uniform in its terminology as to exactly which properties of infinite dimensional groups qualify the group for the prefix lie in lie group. In summary, in in nite dimensions we do not expect a version of chernweil theory. Citeseerx scientific documents that cite the following paper.
Omori, infinite dimensional lie transformation groups, springer lecture notes in mathematics 427 1974 t. In section 2 we give the definition of infinite dimensional manifolds and several examples. For a graded and connected hopf algebra we construct an infinite dimensional. We are offering a phd position in mathematics, in connection with the nwo vidigrant cohomology and representation theory of infinite dimensional lie groups. Infinite dimensional lie theory from the point of view of functional. Glockner, aspects of differential calculus related to infinitedimensional vector bundles and poisson vector spaces, in preparation. Unitary representations of infinite dimensional lie groups of a geometric origin. Remark that convenient vector spaces are exactly those locally convex. Moore 7 on the structure of hopf algebras 1965 with j. Important classes of infinitedimensional lie groups. We think of lie subgroups of these groups as linear lie groups, but we shall only. Lie groups beyond an introduction second edition birkhauser boston basel berlin.
Northholland, 1983 in the case of sequentially complete spaces, carries over without. Universal central extensions of lie groups springerlink. We also characterize the corresponding pro lie algebras in various ways. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Among their topics are spherical varieties and perspectives in representation theory, on toric degenerations of flag varieties, derived categories of quasihereditary algebras and their derived composition series, bounded and semibounded representations of infinite dimensional lie groups, and geometric invariant theory for principal threedimensional subgroups acting on flag varieties. Pdf on the lie subgroups of infinite dimensional lie groups. The main point is the smooth structure on g for infinitedimensional lie groups g which are 1connected, regular, and modeled on a barreled lie. Infinitedimensional lie theory for gauge groups fachbereich. We show that in the presence of suitable commutator estimates, a projective unitary representation of the lie algebra of a connected and simply connec. In particular, it will usually be impossible to tell if these chern classes vanish or not. This lie group is generally called the heisenberg group by 2.
Infinitedimensional representations of compact groups 556 2. We discuss some techniques related to equivariant compactifications of uniform spaces and amenability of topological groups. Character groups of hopf algebras as infinitedimensional. One aspect of the heisenberg algebra story that does not have an analog in the cli. The literature is not entirely uniform in its terminology as to exactly which properties of infinitedimensional. Infinitedimensional lie theory for gauge groups semantic scholar. Infinitedimensional lie groups and algebras in mathematical. It presents some important recent developments in this area. Branching rules of dolbeault cohomology groups over indefinite grassmannian manifolds sekiguchi, hideko, proceedings of the japan academy, series a, mathematical sciences, 2011. A pro lie group is a projective limit of a family of finitedimensional lie groups. In general, there is no consensus on the definition of infinite dimensional lie group, except for the case of banach lie groups 3.
Lie theory more difficult and also richer than the finitedimensional theory. For an infinitedimensional lie group g modeled on a locally convex lie algebra g, we prove that every smooth projective unitary representation of g corresponds to a smooth linear unitary representation of a lie group extension g. Notes on regularity properties of infinitedimensional lie. On the lie subgroups of infinite dimensional lie groups. Let cbe the field of complex numbers and fa vector space over c.
This letter could not influence you to be smarter, however the book infinite dimensional groups and algebras in quantum physics lecture notes in physics monographs, by johnny t. Prolie groups which are infinitedimensional lie groups. Symmetries and innite dimensional lie algebras martin schlichenmaier martin. The main problem discussed in this paper is the existence of central lie group extensions of a connected lie group g which is weakly universal for all abelian lie groups whose identity. In this article character groups of hopf algebras are studied from the perspective of infinitedimensional lie theory. In this case the relation between the lie algebra and the lie group becomes rather subtle, and several results about finite dimensional lie groups no longer hold. Pdf on the lie subgroups of infinite dimensional lie. I general remarks in this talk a loop group lg will mean the group of smooth maps from the circle s i to a compact lie group g.
Many infinite dimensional lie groups of interest can be expressed as a union of an ascending sequence of finite or infinite dimensional lie groups. In this paper we show that a pro lie group g is a lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if g is locally contractible. Remarks on infinite dimensional lie groups milnor, j. Advanced study 1985 51 pages in les houches 1983, proceedings, relativity, groups and topology, ii, 10071057. Functional analytic background for a theory of infinitedimensional. We also characterize the corresponding prolie algebras in various ways. Pdf infinitedimensional groups and their representations. We have chosen to present the unifying ideas of the theory by concentrating on speci. Remarks on actions on compacta by some infinitedimensional. Home remarks on infinite dimensional lie groups information. Pdf dual pairs and infinite dimensional lie algebras. Here we discuss some ways in which equivariant compactifications can be used to study minimal actions and amenability of some infinite dimensional groups. From the group theoretical standpoint, general relativistic mechanics is a more closed system than newtonian mechanics.
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