Showing papers on "Lie group published in 1993"

Natural operations in differential geometry

Abstract: I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index.

Analysis and Geometry on Groups

Abstract: The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical but have little to do with what is described these days as real analysis. Most of the results described in this book have a dual formulation; they have a 'discrete version' related to a finitely generated discrete group, and a continuous version related to a Lie group. The authors chose to centre this book around Lie groups but could quite easily have pushed it in several other directions as it interacts with opetators, and probability theory, as well as with group theory. This book will serve as an excellent basis for graduate courses in Lie groups, Markov chains or potential theory.

Nilpotent Orbits In Semisimple Lie Algebra: An Introduction

Abstract: Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington.

Lectures on Hermite and Laguerre expansions

Abstract: The interplay between analysis on Lie groups and the theory of special functions is well known. This monograph deals with the case of the Heisenberg group and the related expansions in terms of Hermite, special Hermite, and Laguerre functions. The main thrust of the book is to develop a concrete Littlewood-Paley-Stein theory for these expansions and use the theory to prove multiplier theorems. The questions of almost-everywhere and mean convergence of Bochner-Riesz means are also treated. Most of the results in this monograph appear for the first time in book form.

Wess-Zumino-Witten model based on a nonsemisimple group.

Abstract: We present a conformal field theory which describes a homogeneous four dimensional Lorentz-signature space-time. The model is an ungauged Wess-Zumino-Witten model based on a central extension of the Poincar\'e algebra. The central charge of this theory is exactly four, just like four dimensional Minkowski space. The model can be interpreted as a four dimensional monochromatic plane wave. As there are three commuting isometries, other interesting geometries are expected to emerge via O(3,3) duality.

On Poisson homogeneous spaces of Poisson-Lie groups

Abstract: Poisson homogeneous spaces of a Poisson-Lie group G are described in terms of Lagrangian subalgebras of D(g), where D(g) is the double of the Lie bialgebra g corresponding to G.

The Kinematic Formula in Riemannian Homogeneous Spaces

Abstract: Introduction The basic integral formula for submanifolds of a Lie group Poincare's formula in homogeneous spaces Integral invariants of submanifolds of homogeneous spaces, the kinematic formula, and the transfer principle The second fundamental form of an intersection Lemmas and definitions Proof of the kinematic formula and the transfer principle Spaces of constant curvature An algebraic characterization of the polynomials in the Weyl tube formula The Weyl tube formula and the Chern-Federer kinematic formula Appendix: Fibre integrals and the smooth coarea formula References.

Foundations of Lie theory ; Lie transformation groups

Abstract: IFoundations of Lie Theory- 1 Basic Notions- 1 Lie Groups, Subgroups and Homomorphisms- 11 Definition of a Lie Group- 12 Lie Subgroups- 13 Homomorphisms of Lie Groups- 14 Linear Representations of Lie Groups- 15 Local Lie Groups- 2 Actions of Lie Groups- 21 Definition of an Action- 22 Orbits and Stabilizers- 23 Images and Kernels of Homomorphisms- 24 Orbits of Compact Lie Groups- 3 Coset Manifolds and Quotients of Lie Groups- 31 Coset Manifolds- 32 Lie Quotient Groups- 33 The Transitive Action Theorem and the Epimorphism Theorem- 34 The Pre-image of a Lie Group Under a Homomorphism- 35 Semidirect Products of Lie Groups- 4 Connectedness and Simply-connectedness of Lie Groups- 41 Connected Components of a Lie Group- 42 Investigation of Connectedness of the Classical Lie Groups- 43 Covering Homomorphisms- 44 The Universal Covering Lie Group- 45 Investigation of Simply-connectedness of the Classical Lie Groups- 2 The Relation Between Lie Groups and Lie Algebras- 1 The Lie Functor- 11 The Tangent Algebra of a Lie Group- 12 Vector Fields on a Lie Group- 13 The Differential of a Homomorphism of Lie Groups- 14 The Differential of an Action of a Lie Group- 15 The Tangent Algebra of a Stabilizer- 16 The Adjoint Representation- 2 Integration of Homomorphisms of Lie Algebras- 21 The Differential Equation of a Path in a Lie Group- 22 The Uniqueness Theorem- 23 Virtual Lie Subgroups- 24 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra- 25 Deformations of Paths in Lie Groups- 26 The Existence Theorem- 27 Abelian Lie Groups- 3 The Exponential Map- 31 One-Parameter Subgroups- 32 Definition and Basic Properties of the Exponential Map- 33 The Differential of the Exponential Map- 34 The Exponential Map in the Full Linear Group- 35 Cartan's Theorem- 36 The Subgroup of Fixed Points of an Automorphism of a Lie Group- 4 Automorphisms and Derivations- 41 The Group of Automorphisms- 42 The Algebra of Derivations- 43 The Tangent Algebra of a Semi-Direct Product of Lie Groups- 5 The Commutator Subgroup and the Radical- 51 The Commutator Subgroup- 52 The Maltsev Closure- 53 The Structure of Virtual Lie Subgroups- 54 Mutual Commutator Subgroups- 55 Solvable Lie Groups- 56 The Radical- 57 Nilpotent Lie Groups- 3 The Universal Enveloping Algebra- 1 The Simplest Properties of Universal Enveloping Algebras- 11 Definition and Construction- 12 The Poincare-Birkhoff-Witt Theorem- 13 Symmetrization- 14 The Center of the Universal Enveloping Algebra- 15 The Skew-Field of Fractions of the Universal Enveloping Algebra- 2 Bialgebras Associated with Lie Algebras and Lie Groups- 21 Bialgebras- 22 Right Invariant Differential Operators on a Lie Group- 23 Bialgebras Associated with a Lie Group- 3 The Campbell-Hausdorff Formula- 31 Free Lie Algebras- 32 The Campbell-Hausdorff Series- 33 Convergence of the Campbell-Hausdorff Series- 4 Generalizations of Lie Groups- 1 Lie Groups over Complete Valued Fields- 11 Valued Fields- 12 Basic Definitions and Examples- 13 Actions of Lie Groups- 14 Standard Lie Groups over a Non-archimedean Field- 15 Tangent Algebras of Lie Groups- 2 Formal Groups- 21 Definition and Simplest Properties- 22 The Tangent Algebra of a Formal Group- 23 The Bialgebra Associated with a Formal Group- 3 Infinite-Dimensional Lie Groups- 31 Banach Lie Groups- 32 The Correspondence Between Banach Lie Groups and Banach Lie Algebras- 33 Actions of Banach Lie Groups on Finite-Dimensional Manifolds- 34 Lie-Frechet Groups- 35 ILB- and ILH-Lie Groups- 4 Lie Groups and Topological Groups- 41 Continuous Homomorphisms of Lie Groups- 42 Hilbert's 5-th Problem- 5 Analytic Loops- 51 Basic Definitions and Examples- 52 The Tangent Algebra of an Analytic Loop- 53 The Tangent Algebra of a Diassociative Loop- 54 The Tangent Algebra of a Bol Loop- References- II Lie Transformation Groups- 1 Lie Group Actions on Manifolds- 1 Introductory Concepts- 11 Basic Definitions- 12 Some Examples and Special Cases- 13 Local Actions- 14 Orbits and Stabilizers- 15 Representation in the Space of Functions- 2 Infinitesimal Study of Actions- 21 Flows and Vector Fields- 22 Infinitesimal Description of Actions and Morphisms- 23 Existence Theorems- 24 Groups of Automorphisms of Certain Geometric Structures- 3 Fibre Bundles- 31 Fibre Bundles with a Structure Group- 32 Examples of Fibre Bundles- 33 G-bundles- 34 Induced Bundles and the Classification Theorem- 2 Transitive Actions- 1 Group Models- 11 Definitions and Examples- 12 Basic Problems- 13 The Group of Automorphisms- 14 Primitive Actions- 2 Some Facts Concerning Topology of Homogeneous Spaces- 21 Covering Spaces- 22 Real Cohomology of Lie Groups- 23 Subgroups with Maximal Exponent in Simple Lie Groups- 24 Some Homotopy Invariants of Homogeneous Spaces- 3 Homogeneous Bundles- 31 Invariant Sections and Classification of Homogeneous Bundles- 32 Homogeneous Vector Bundles The Frobenius Duality- 33 The Linear Isotropy Representation and Invariant Vector Fields- 34 Invariant A-structures- 35 Invariant Integration- 36 Karpelevich-Mostow Bundles- 4 Inclusions Among Transitive Actions- 41 Reductions of Transitive Actions and Factorization of Groups- 42 The Natural Enlargement of an Action- 43 Some Inclusions Among Transitive Actions on Spheres- 44 Factorizations of Lie Groups and Lie Algebras- 45 Factorizations of Compact Lie Groups- 46 Compact Enlargements of Transitive Actions of Simple Lie Groups- 47 Groups of Isometries of Riemannian Homogeneous Spaces of Simple Compact Lie Groups- 48 Groups of Automorphisms of Simply Connected Homogeneous Compact Complex Manifolds- 3 Actions of Compact Lie Groups- 1 The General Theory of Compact Lie Transformation Groups- 11 Proper Actions- 12 Existence of Slices- 13 Two Fiberings of an Equi-orbital G-space- 14 Principal Orbits- 15 Orbit Structure- 16 Linearization of Actions- 17 Lifting of Actions- 2 Invariants and Almost-Invariants- 21 Applications of Invariant Integration- 22 Finiteness Theorems for Invariants- 23 Finiteness Theorems for Almost Invariants- 3 Applications to Homogeneous Spaces of Reductive Groups- 31 Complexification of Homogeneous Spaces- 32 Factorization of Reductive Algebraic Groups and Lie Algebras- 4 Homogeneous Spaces of Nilpotent and Solvable Groups- 1 Nilmanifolds- 11 Examples of Nilmanifolds- 12 Topology of Arbitrary Nilmanifolds- 13 Structure of Compact Nilmanifolds- 14 Compact Nilmanifolds as Towers of Principal Bundles with Fibre T1- 2 Solvmanifolds- 21 Examples of Solvmanifolds- 22 Solvmanifolds and Vector Bundles- 23 Compact Solvmanifolds (The Structure Theorem)- 24 The Fundamental Group of a Solvmanifold- 25 The Tangent Bundle of a Compact Solvmanifold- 26 Transitive Actions of Lie Groups on Compact Solvmanifolds- 27 The Case of Discrete Stabilizers- 28 Homogeneous Spaces of Solvable Lie Groups of Type (I)- 29 Complex Compact Solvmanifolds- 5 Compact Homogeneous Spaces- 1 Uniform Subgroups- 11 Algebraic Uniform Subgroups- 12 Tits Bundles- 13 Uniform Subgroups of Semi-simple Lie Groups- 14 Connected Uniform Subgroups- 15 Reductions of Transitive Actions of Reductive Lie Groups- 2 Transitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups- 21 Three Lemmas on Transitive Actions- 22 Radical Enlargements- 23 A Sufficient Condition for the Radical to be Abelian- 24 Passage from Compact Groups to Non-Compact Semi-simple Groups- 25 Compact Homogeneous Spaces of Rank 1- 26 Transitive Actions of Non-Compact Lie Groups on Spheres- 27 Existence of Maximal and Largest Enlargements- 3 The Natural Bundle- 31 Orbits of the Action of a Maximal Compact Subgroup- 32 Construction of the Natural Bundle and Its Properties- 33 Some Examples of Natural Bundles- 34 On the Uniqueness of the Natural Bundle- 35 The Case of Low Dimension of Fibre and Basis- 4 The Structure Bundle- 41 Regular Transitive Actions of Lie Groups- 42 The Structure of the Base of the Natural Bundle- 43 Some Examples of Structure Bundles- 5 The Fundamental Group- 51 On the Concept of Commensurability of Groups- 52 Embedding of the Fundamental Group in a Lie Group- 53 Solvable and Semi-simple Components- 54 Cohomological Dimension- 55 The Euler Characteristic- 56 The Number of Ends- 6 Some Classes of Compact Homogeneous Spaces- 61 Three Components of a Compact Homogeneous Space and the Case when Two of them Are Trivial- 62 The Case of One Trivial Component- 7 Aspherical Compact Homogeneous Spaces- 71 Group Models of Aspherical Compact Homogeneous Spaces- 72 On the Fundamental Group- 8 Semi-simple Compact Homogeneous Spaces- 81 Transitivity of a Semi-simple Subgroup- 82 The Fundamental Group- 83 On the Fibre of the Natural Bundle- 9 Solvable Compact Homogeneous Spaces- 91 Properties of the Natural Bundle- 92 Elementary Solvable Homogeneous Spaces- 10 Compact Homogeneous Spaces with Discrete Stabilizers- 6 Actions of Lie Groups on Low-dimensional Manifolds- 1 Classification of Local Actions- 11 Notes on Local Actions- 12 Classification of Local Actions of Lie Groups on ?1, ?1- 13 Classification of Local Actions of Lie Groups on ?2 and ?2- 2 Homogeneous Spaces of Dimension ?3- 21 One-dimensional Homogeneous Spaces- 22 Two-dimensional Homogeneous Spaces (Homogeneous Surfaces)- 23 Three-dimensional Manifolds- 3 Compact Homogeneous Manifolds of Low Dimension- 31 On Four-dimensional Compact Homogeneous Manifolds- 32 Compact Homogeneous Manifolds of Dimension ?6- 33 On Compact Homogeneous Manifolds of Dimension ?7- References

An introduction to noncommutative differential geometry on quantum groups

Abstract: We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case (q→1 limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan-Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group GLq(2) is given in detail. The softening of a quantum group is considered, and we introduce q curvatures satisfying q Bianchi identities, a basic ingredient for the construction of q gravity and q gauge theories.

Symmetries and conditional symmetries of differential difference equations

Abstract: Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations, in particular, to differential equations of the delay type. The concept of conditional symmetries is extended from purely differential to differential‐difference equations and shown to incorporate Backlund transformations.

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Abstract: Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU q (g). They have the same FRT generatorsl ± but a matrix braided-coproductΔL=L⊗L, whereL=l + Sl −, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U q (sl 2)) (also known as the “quantum Lorentz group”) is the semidirect product as an algebra of two copies ofU q (sl 2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.

Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n))

Abstract: Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite‐dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.

Random Walks in Weyl Chambers and the Decomposition of Tensor Powers

Abstract: We consider a class of random walks on a lattice, introduced by Gessel and Zeilberger, for which the reflection principle can be used to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We prove three independent results about such “reflectable walks”: first, a classification of all such walkss second, many determinant formulas for walk numbers and their generating functionss third, an equality between the walk numbers and the multiplicities of irreducibles in the kth tensor power of certain Lie group representations associated to the walk types. Our results apply to the defining representations of the classical groups, as well as some spin representations of the orthogonal groups.

Asymptotic expansion of stochastic flows

Abstract: We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].

Lie Groups and Solutions of Nonlinear Partial Differential Equations

Abstract: The application of local Lie point transformation groups to the solution of partial differential equations is reviewed. The method of symmetry reduction is presented as an algorithm. Included is the construction of group invariant solutions, partially invariant solutions and also the use of conditional symmetries. The emphasis in on recent developments, including the use of computer algebra. Many examples and applications are treated.

Quantizing sl(n) solitons and the Hecke algebra

Abstract: The problem of quantizing a class of two-dimensional integrable quantum field theories is considered. The classical equations of the theory are the complex sl(n) affine Toda equations which admit soliton solutions with real masses. The classical scattering theory of the solitons is developed using Hirota’s solution techniques. A form for the soliton S matrix is proposed based on the constraints of S matrix theory, integrability and the requirement that the semiclassical limit is consistent with the semiclassical WKB quantization of the classical scattering theory. The proposed S matrix is an intertwiner of the quantum group associated to sl(n), where the deformation parameter is a function of the coupling constant. It is further shown that the S matrix describes a nonunitary theory, which reflects the fact that the classical Hamiltonian is complex. The spectrum of the theory is found to consist of the basic solitons, excited (or ‘breathing’) solitons, scalar states (or breathers) and solitons transforming in nonfundamental representations. For some region of coupling constant space only the original solitons are in the spectrum and so the S matrix is complete, in addition arguments are presented which indicate that in a more restricted region the theory is actually unitary. It is also noted that the construction of the S matrix is valid for any representation of the Hecke algebra, allowing the definition of restricted S matrices, which lie in the unitary and complete region.

RiemannianG-manifold with one-dimensional orbit space

Abstract: Cohomogeneity one RiemannianG-manifolds (i.e. Riemannian manifolds with a groupG of isometries having an orbit of codimension one) are studied. A description of such manifolds (up to some normal equivalence) is given in terms of Lie subgroups of Lie groupG. The twist of a geodesic normal to all orbits is defined as the number of intersections with a singular orbit. It is equal to the order of some Weyl group, associated with theG-manifold. Some results about possible values of the twist are obtained.

Spectral invariance of dense subalgebras of operator algebras

Abstract: We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G ⋊ A is spectral invariant in the C*-crossed product G ⋊ B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one of these. Our theorem gives the spectral invariance of G ⋊ A if A is the set of C∞-vectors for the action of G on B, or if B = C0 (M), and A is a set of G-differentiable Schwartz functions on M. This gives many examples of spectral invariant dense subalgebras for the C*-algebras associated with dynamical systems. We also obtain relevant results about exact sequences, subalgebras, tensoring by smooth compact operators, and strong spectral invariance in L1 (G, B).

Invariant control sets on flag manifolds

Abstract: LetG be a semisimple Lie group and letS ⊂G be a subsemigroup with nonempty interior. In this paper we study invariant control sets for the action ofS on homogeneous spaces ofG. These sets on the boundary manifolds of the group are characterized in terms of the semisimple elements contained in intS. From this characterization a result on controllability of control systems on semisimple Lie groups is derived. Invariant control sets for the action of S on the boundaries of larger groups¯G withG ⊂¯G are also studied. This latter case includes the action ofS on the projective space and on the flag manifolds.

Group actions and deformations for harmonic maps

Abstract: From the theory of integrable systems it is known that harmonic maps from a Riemann surface to a Lie group may be studied by infinite dimensional methods (cf. [ZM],[ZS]). This was clarified considerably by the papers [Uh],[Se], especially in the case of maps from the Riemann sphere S to a unitary group Un. The basic connection with infinite dimensional methods is the correspondence between harmonic maps S −→ G and “extended solutions” S −→ ΩG, where G is any compact Lie group and ΩG is its (based) loop group. In [Uh] this was used in two ways (in the case G = Un):

Lévy flows on manifolds and Lévy processes on Lie groups

Abstract: The main concern of this paper is to construct stochastic flows of diffeomorphisms of manifolds by solving stochastic differential equations (SDE's) driven by Lévy processes. In two earlier papers of Fujiwara-Kunita ([2]) and Fujiwara ([1]), existence and uniqueness of the solutions of such equations were established in th e first place on Rd and in the second p lace when the m anifold (M ) was com pact. Herein we will not restrict ourselves to compact manifolds and will aim to find some natural classes of SDE's whose solutions take values in the diffeomorphism group o f th e m an ifo ld . In fact we will aim to generalize the well known result fo r flows driven by Brownian m otion wherein the solution consists of diffeomorphisms (almost surely) provided each o f th e vector fields driving the equation is deterministically complete and the Lie algebra which they generate is finite dimensional (see [6 ] Theorem 4.8.7). We note that in [1] and [2] it was shown that the solutions of the stochastic differential equations described therein define Lévy processes (i.e. cddldg processes with independent increments) o n G , a n d G'T respectively where G.,. (G7) is the topological semigroup comprising continuous maps (C m maps) from Rd o r M into itself. Furthermore, it is shown in [5] that under some additional conditions, the so lu tion in [2] defines a L évy process o n Gm where Gm is th e topological group of Cm-diffeomorphisms of Rd . However the la tter argum ent cannot be applied in this case since it depends critically on the global properties of Euclidean space. Hence we develop a completely different method. A m a jo r difference between this paper a n d its predecessors is that we restrict our Lévy process driving the SDE to possess finitely many degrees of freedom so that in particular the Poisson random measure component of the process is itself defined on the finite dimensional manifold N . We construct two distinct classes of Lévy flows in this paper which are obtained a s follows. (i) N = Rd a n d th e vector fields driving the SD E satisfy the condition on the Lie algebra described in the first paragraph above. (ii) N is a finite dimensional Lie group and the vector fields driving the SDE belong to the L ie algebra of N.

Optimal Control and Poisson Reduction

Abstract: : In this paper, the author has worked out explicitly the Poisson reduction of certain G-invariant optimal control problems on Lie groups. The approach presented yields an algorithm for constructing regular extremals.

Dense m-convex Frechet Subalgebras of Operator Algebra Crossed Products by Lie Groups

Abstract: Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily connected, which acts on both A and B by *-automorphisms, and let σ be a sub-polynomial function from G to the nonnegative real numbers. If σ and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G ⋊σ A of the crossed product L1 (G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in σ. We give conditions on σ and the action of G on A which imply the m-convexity of the dense subalgebra G ⋊σ A. A locally convex algebra is said to be m-convex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Frechet algebra, and is useful in modern operator theory. If G acts as a transformation group on a locally compact space M, we develop a class of dense subalgebras for the crossed product L1 (G, C0 (M)), where C0 (M) denotes the continuous functions on M vanishing at infinity with the sup norm topology. We define Schwartz functions S (M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense Frechet *-subalgebra G ⋊σ S (M) of rapidly vanishing, G-differentiable functions from G to S (M). If the reciprocal of σ is in Lp (G) for some p, we prove that our group algebras Sσ (G) are nuclear Frechet spaces, and that G ⋊σ A is the projective completion .