Showing papers on "Representation theory published in 1974"

Representation Theory of Artin Algebras I

Abstract: This is the first of a series of papers dealing with the representation theory of artin algebras, where by an artin algebra we mean an artin ring having the property that its center is an artin ring and λ is a finitely generated module over its center. The over all purpose of this paper is to develop terminology and background material which will be used in the rest of the papers in the series. While it is undoubtedly true that much of this material can be found in the literature or easily deduced from results already in the literature, the particular development presented here appears to be new and is especially well suited as a foundation for the papers to come.

Conjugacy Classes in Algebraic Groups

Abstract: Affine algebraic varieties, affine algebraic groups and their orbits.- First Part: Jordan decompositions, unipotent and diagonalizable groups.- Second Part: Quotients and solvable groups.- Reductive and semisimple algebraic groups, regular and subregular elements.

Nonrelativistic current algebra in the N / V limit

Abstract: The case of a noninteracting infinite Bose gas at zero temperature is studied in the formalism of local current algebras, using the representation theory of nuclear Lie groups. The class of representations describing such a system is obtained by taking an ``N / V limit'' of the finite case. These representations can also be determined uniquely from the solutions of a functional differential equation, which follows in turn from a condition on the ground state vector. Finally a system of functional differential equations is formulated for a theory with interactions, using a proposed definition of indefinite functional integration.

Finite dimensional representations of algebras

Abstract: Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.

Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxx −c/x2 U = 0

Abstract: A detailed study of the group of symmetries of the time‐dependent free particle Schrodinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well‐defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x2 is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrodinger equation, several of which are new.

Infinite dimensional Lie transformations groups

Abstract: General theory of strong ILB-Lie groups and subgroups.- Groups of diffeomorphisms.- Basic theorems I.- Vector bundle over strong ILB-Lie groups.- Review of the smooth extension theorem and a remark on elliptic operators.- Basic theorems II (Frobenius theorem).- Frobenius theorem on strong ILB-Lie groups.- Miscellaneous examples.- Primitive transformation groups.- Lie algebras of vector fields.- Linear groups and groups of diffeomorphisms.

Arithmetic Properties of Discrete Subgroups

Abstract: That the factor space of a semisimple Lie group by an arithmetic subgroup has finite volume with respect to Haar measure is well known. In this paper we study results related to the converse of this theorem. In particular, under some rather weak assumptions on a semisimple Lie group G we prove that every discrete subgroup of G with a non-compact factor space of finite volume that satisfies some natural irreducibility conditions, is an arithmetic subgroup of G. In this paper we also study various results from the theory of algebraic groups and their arithmetic and discrete subgroups. In the proof of one theorem we use a construction from representation theory that is of independent interest. At the end we state some unsolved problems in the theory of discrete subgroups.

Lie theory and separation of variables. 4. The groups SO (2,1) and SO (3)

Abstract: Winternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid x02−x12−x22=1 separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one‐variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well‐known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lame functions. We show that there is virtually a perfect correspondence between the known theory of the Lame functions and the representation theory of SO (2,1) and SO (3).

Lie Theory and Separation of Variables. I: Parabolic Cylinder Coordinates

Abstract: Winternitz and co-workers have characterized the parabolic cylinder function solutions of the reduced wave equation in two variables as eigenfunctions of a quadratic operator $E = MP_2 + P_2 M$ in the enveloping algebra of the Lie algebra of the Euclidean group in the plane. Here we study the representation theory of the Euclidean and pseudo-Euclidean groups in an E-basis and use the results to derive a number of new addition and expansion theorems for products of parabolic cylinder functions.

On the K-theory of the loop space of a Lie group

Abstract: The thesis studies the K-theory of the loop space of compact Lie groups. There are three independent chapters. The first chapter is devoted to proving the non-convergence of a K-theory Eilenberg-Maclane spectral sequence for the path space fibration on a Lie group with torsion in its cohomology. The second chapter generalises a method of Bott. We obtain an algebraic expression for K*(ΩG) in terms of the" representation theory of the group G. Explicit results are obtained for the unitary groups and the first symplectic group. The third chapter develops and extends a method due to T. Petrie to find the Hopf algebra structure of the complex bordism of .ΩG. The Conner-Floyd isomorphism is used to obtain explicit results for K*(ΩG) for a larger class of groups than in chapter two.

Noncompact Lie‐algebraic approach to the unitary representations of SU∼(1,1): Role of the confluent hypergeometric equation

Abstract: The continuous unitary irreducible representations of the covering group SU∼ (1,1) of SU(1,1) are studied using the self‐adjoint representations of the Lie algebra in the case that one of the noncompact generators is diagonal. The study can be carried out for all the representations simultaneously and is shown to reduce to a study of the self‐adjointness of the compact element of the Lie algebra, which in this basis turns out to be the confluent hypergeometric operator. Several basic results, such as the classification of the representations, and a formula for the transformation coefficients from the compact to the noncompact basis which is valid for all representations, emerge quite simply.

The category of generalized Lie groups

Abstract: We consider the category r of generalized Lie groups. A generalized Lie group is a topological group G such that the set LG = Hom (R, G) of continuous homomorphisms from the reals R into G has certain Lie algebra and locally convex topological vector space structures. The full subcategory rr of r-bounded (r positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in r. Various properties of generalized Lie groups G and their locally convex topological Lie algebras LG = Hom (R, G) are investigated.

Introduction to classification theory of algebraic varieties and compact complex spaces

Abstract: Classification theory of algebraic varieties and compact complex spaces, by Kenji Ueno, Lecture Notes in Math., no. 439, Springer-Verlag, Berlin, Heidelberg, New York, 1975, 278 + xix pp., $12.10. Looking for a research area in which you can start at the ground floor? This is it—the classification of varieties in dimensions three and higher. Of course the prerequisites might sound a little stiff: algebraic geometry, complex analytic geometry, and the "classical" classification theory in dimensions one and two, but in reality it's not as bad as one might fear. The author of the text under review gave an informal course on the subject at the University of Mannheim in 1972, and the lecture notes (by P. Cherenack) form the basis of the text. At the very least this book is a way to take a peek at what is going on in this new field, and maybe even it is a way to get into it. The first pleasant surprise the neophyte encounters in the study of geometry is that the two main categories in which geometers work-the analytic category and the algebraic category-actually have a very large overlap. Thus studying one category allows one to absorb "by osmosis" results in the other. The terminology is sometimes different; for example, here is a short, rough, transliteration guide:

Lie Theory and Separation of Variables. II: Parabolic Coordinates

Abstract: Winternitz and coworkers have characterized those solutions of the equation $(\Delta _3 + \omega ^2) f(x) = 0$ which are expressible as products of functions of the paraboloid of revolution, as simultaneous eigenfunctions of the commuting quadratic operators \[E = J_1 P_2 , + P_2 J_1 , - P_1 J_2 - J_2 P_1 ,J_3^2 \] in the enveloping algebra of the Lie algebra of the Euclidean group in three-space $E(3)$. Here we study the representation theory of the real and complex Euclidean groups in an $E - J_3^2 $ basis and use the results to derive some addition and expansion theorems for parabolic functions which simplify and in some cases extend identities due to Buchholz and Hochstadt. We also give the decomposition of the quasi-regular representation of $E(3)$ in an $E - J_3^2 $ basis.

Duality theories for metabelian Lie algebras

Abstract: This paper is concerned with duality theories for metabelian (2-step nilpotent) Lie algebras. A duality theory associates to each metabelian 2 Lie algebra N with cod N = g, another such algebra ND satisfying (ND)D N, N1 -N2 if and only if (N)DD (N2)D, and if dim N = g + p then dim ND= g + (2) p. The obvious benefit of such a theory lies in its reduction of the classification problem. Introduction. The purpose of this paper is to determine the "reasonable" duality theories for metabelian Lie algebras over an algebraically closed field of characteristic zero. If N is a metabelian Lie algebra (i.e. N3 = -; we are including the abelian Lie algebras as "degenerate" metabelian Lie algebras although this is not standard) such that cod N2 = g, a duality theory associates to N another such algebra ND, the dual, satisfying the formal properties 1. (ND)D N 2. N1 N2 if and only if (Nd)D-(N 2)D' and 3. if dim N = g + p, then dim ND = g + (g) P. Two such theories were developed independently by Scheuneman [4] and myself [1]. Their obvious benefit is in the reduction of the classification problem. In this paper, I define an algebraic duality theory to be an association N -p ND satisfying conditions a little stronger than the three above, plus a requirement, roughly speaking, that the structure constants of ND depend algebraically on those of N. These axioms, which are satisfied by both Scheuneman's and my own duality, result in a uniqueness theorem, hence the identity of the two theories just mentioned. It appeared at first that there were two algebraic duality theories. I am, however, indebted to the referee for detecting an error in my original arguments and for giving a proof of the uniqueness. A homological approach is taken to the subject by Leger and Luks in [3]. They also arrive at a uniqueness theorem. The main difference between these two papers appears to be their assumption that a duality D is a contravariant functor. Hence, for any homomorphism 0: N -. M of metabelian Lie algebras, there is a homomorphism OD: MD -. ND. We make no such requirement here. Due to the generator-relation view given to the classification of metabelian Received by the editors April 17, 1972. AMS (MOS) subject classifications (1970). Primary 17B30. Copyright

Functional analysis and its applications : international conference, Madras, 1973

Abstract: Non-abelian pontryagin duality.- The topological dual, the algebraic dual and Radon-Nikodym derivatives.- Segal algebras and dense ideals in Banach algebras.- Field approximation and free approximation for differential equations.- Linear interpolation and linear extension of functions.- Wave front sets and hypoelliptic operators.- Harmonic analysis in the complex domain and applications in the theory of analytical and infinitely differentiable functions.- Muntz-szasz theorem with integral coefficients I.- Inductive limits of Banach spaces and complex analysis.- Representation of nonlinear operators with the hammerstein property.- On polynomial approximation with respect to general weights.- Determination of conformal modules of ring domains and quadrilaterals.- Solovay's axion and functional analysis.- Q - Uniform algebras and operator theory.- On polynomials with a prescribed zero.- A priori inequalities for systems of partial differential equations.- Recent results on Segal algebras.- Measurability of lattice operations in a cone.- On some nonlinear elliptic boundary value problems.- Quasicomplemented Banach algebras.- On the (Lp, Lp) multipliers.- Heredity in metric projections.- Multipliers on weighted spaces.- On properties of traces of functions belonging to weight spaces.- Fundamental solutions of hyperbolic differential equations.- Non-self-conjugate differential dirac operators expansion in eigenfunctions through the whole axis.- Topological algebras in several complex variables.- Spectra of composition operators on C[0,1].- Linear functionals on vector valued kothe spaces.- A general view on unitary dilations.- Quantization in Hamiltonian particle mechanics.- The lebesgue constants for polynomial interpolation.- Approximation theorems for polynomial spline operators.- Characterization of the barrelled, d-barrelled and ?-barrelled spaces of continuous functions.- Cardinal spline interpolation and the exponential Euler splines.- Approximation of analytic functions in Hausdorff metric.- Invariant means and almost convergence in non-Archimedean analysis.- Parameasures and multipliers of Segal algebras.- Segal algebras of Beurling type.- Splines in Hilbert spaces.- A survey of v-integral representation theory for operators on function spaces including the topological vector space setting.- Isotone measures, 1948-1973.