Showing papers on "Representation theory published in 1979"

Transformation groups and representation theory

Abstract: The Burnside ring of finite G-sets.- The J-homomorphism and quadratic forms.- ?-rings.- Permutation representations.- The Burnside-ring of a compact Lie group.- Induction theory.- Equivariant homology and cohomology.- Equivariant homotopy theory.- Homotopy equivalent group representations.- Geometric modules over the Burnside ring.- Homotopy-equivalent stable G-vector bundles.

Representations of finite groups of Lie type

Abstract: The representation theory of a group G over the field of complex numbers involves two problems: first, the construction of the irreducible representations of G; and second, the problem of expressing each suitably restricted complex valued function on G, as a linear combination (or a limit of linear combinations), of the coefficients of the irreducible representations. For example, if G is the additive group of real numbers mod 1 (the one-dimensional torus), one considers integrable functions on G, or what is the same thing, integrable periodic functions of period 1 on the additive group of real numbers. In this case the irreducible representations of G are given by the exponential functions x -» e, where k is an integer, and are the continuous homomorphisms from G into the multiplicative group of complex numbers. The expression of an integrable function in terms of the irreducible representations {e} is the Fourier expansion of/,

Commuting varieties of semisimple Lie algebras and algebraic groups

Abstract: © Foundation Compositio Mathematica, 1979, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Duality theory for covariant systems

Abstract: If (A, p, G) is a covariant system over a locally compact group G, i.e. p is a homomorphism from G into the group of *-automorphisms of an operator algebra A, there is a new operator algebra W called the covariance algebra associated with (A, p, G). If A is a von Neumann algebra and p is a-weakly continuous, W is defined such that it is a von Neumann algebra. If A is a C*-algebra and p is norm-continuous W will be a C*-algebra. The following problems are studied in these two different settings: 1. If W is a covariance algebra, how do we recover A and p? 2. When is an operator algebra W the covariance algebra for some covariant system over a given locally compact group G? Introduction. If G is a locally compact group and p: G -Aut(A) is a continuous homomorphism of G into the group of *-automorphisms of an operator algebra A, the triple (A, p, G) is called a covariant system. (A more precise definition is given in Chapters 2 and 3.) This notation was introduced by Doplicher, Kastler and Robinson in 1966, [10], but already Murray and von Neumann considered special cases with A abelian and G discrete in constructing the first non-type I factors. Covariant systems have turned out to be very interesting objects both in theoretical physics and in mathematics. With a representation of (A, p, G) we shall mean a pair (S, U) consisting of a unitary representation U of G and a *-representation S of A with S and U operating over the same Hilbert space such that Spx(a)' UxSaUxI fora&A,xe G. Doplicher, Kastler and Robinson showed that the representation theory of (A, p, G) was essentially the same as that of a certain operator algebra 9l called the covariance algebra of (A, p, G). The representation theory of W has been extensively studied by M. Takesaki in [22], G. Zeller-Meier in [30] and E. G. Effros and F. Hahn in [11] among others, and the covariance algebras provide us with a rich variety of examples of operator algebras. For instance many examples of factors are obtained this way (cf. [20, Chapter 4.2] for some), and A. Connes and M. Takesaki have recently shown that all type III Received by the editors March 13, 1975. AMS (MOS) subject classifications (1970). Primary Primary 46L05, 46L10.

Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras

Abstract: We present the commutation and anticommutation relations, satisfied by the generators of the graded general linear, special linear and orthosymplectic Lie algebras, in canonical two‐index matrix form. Tensor operators are constructed in the enveloping algebra, including powers of the matrix of generators. Traces of the latter are shown to yield a sequence of Casimir invariants. The transformation properties of vector operators under these algebras are also exhibited. The eigenvalues of the quadratic Casimir invariants are given for the irreducible representations of ggl(m ‖ n), gsl(m ‖ n), and osp(m ‖ n) in terms of the highest‐weight vector. In such representations, characteristic polynomial identities of order (m+n), satisfied by the matrix of generators, are obtained in factorized form. These are used in each case to determine the number of independent Casimir invariants of the trace form.

Dynamics on the Group Manifold and Path Integral

Abstract: Classical and quantal dynamics on the compact simple Lie group and on a sphere of arbitrary dimensionality are considered. The accuracy of the semiclassical approximation for Green's functions is discussed. Various path integral representations of Green's functions are presented. The special features of these representations due to the compactness and curvature are analysed. Basic results of the theory of Lie algebras and Lie groups used in the main text are presented in the Appendix.

On the springer representations of the weyl groups of classical algebraic groups

Abstract: (1979). On the springer representations of the weyl groups of classical algebraic groups. Communications in Algebra: Vol. 7, No. 16, pp. 1713-1745.

Dimensions of irreducible representations of the classical Lie groups

Abstract: Formulae are derived expressing the dimensions of each irreducible representation of each of the classical Lie groups of transformations in an N-dimensional space as a factored polynomial in N divided by a product of hook length factors.

Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups

Abstract: We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

On algebraic groups defined by Jordan pairs

Abstract: Let G be an algebraic group over a field k, and let ψ be an action of the multiplicative group km of k on G by automorphisms. We say ψ is an elementary action if it has only the weights 0, ±1; more precisely, if there exist subgroups H, U+, U- of G such that (i) H is fixed under ψ, (ii) U+ and U+ are vector groups and (iii) Ω = U+. H . U+ is open in G, and (iv) G is generated by H, U+, U+ . This situation is characteristic for the complexifications of the automorphism groups of bounded symmetric domains (see, e.g., [9, 16]). A typical example is G = GLnwith (matrices being decomposed into 4 blocks) ψ given by

On Rossmann's Character Formula for Discrete Series.

Abstract: In a recent article in this journal, W. Rossmann [3] has proved a formula conjectured by Kirillov connecting characters of the representations of semisimple Lie groups occurring in the Plancherel formula with Fourier transforms of the measures on orbits in the coadjoint representation. Rossmann's formula has considerable conceptual importance, as it established, via the orbit method, the connection between the representation theory of general Lie groups and the cabalistic study of representations of semi-simple Lie groups. I give here a simple proof of Rossmann's basic theorem, which relates the Fourier transforms on g and on a Cartan subalgebra of compact type. 1. Let V be a real vector space with a non-degenerate symmetric form B. I consider the multiplication operator m 8 given by (mB.f)(x)=B(x,x)f(x), and AB the c~176 Laplace ~176 (AB" f)(x)=(B (~--x' ~x)" f )(x)" i ing the harmonic oscillator i(mB--AB) as JB, [if V is one dimensional, JB=i [.X 2

Branching theorems for semisimple Lie groups of real rank one

Abstract: L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Characteristic cobordism classes and topological applications of the theories of one-valued and two-valued formal groups

Abstract: This survey is devoted to a detailed exposition of the relations between algebraic topology and the theory of formal Lie groups. The focus of attention is the theory of characteristic classes of vector bundles lying at the basis of these relations.

Geometrical theory of contractions of groups and representations

Abstract: The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.

Differential algebraic Lie algebras

Abstract: A class of infinite-dimensional Lie algebras over the field SC of constants of a universal differential field G1i is studied. The simplest case, defined by homogeneous linear differential equations, is analyzed in detail, and those with underlying set c1t x c1t are classified. Introduction. Let Q1t be a universal differential field of characteristic zero with set A of commuting derivation operators and field S(; of constants. We study a class of Lie algebras over S(; called differential algebraic. A differential algebraic Lie algebra g is, in general, infinite-dimensional but has defined on it an additional structure that gives it a tractability it might not otherwise have. We require the additive group g+ to be a differential algebraic group relative to the universe Q1t (roughly speaking, a group object in the category of differential algebraic sets in the sense of Kolchin and Ritt). We also require of the Lie product and scalar multiplication operations that they be morphisms of differential algebraic sets. The Lie algebra g thus inherits, in particular, the finite differential dimensionality of its additive group. Throughout, we will use the prefix "A-" (or "8-" if Q1t is an ordinary differential field with derivation operator 8) in place of "differential algebraic" and "differential rational." The primed letter a' will always stand for 3a. If i > 0, a(i) denotes dia. A-Lie algebras arise naturally in the development of a suitable Lie theory for A-groups. If G is a connected A-group and Q1t is the differential field of A-functions on G, a differential derivation on G is a derivation D of Qt over Q1t such that D o 8 = 8 o D (3 E A). G acts (through right translations) on the set of differential derivations. The set of differential derivations on G that are invariant under this action is readily observed to be a Lie algebra over the field SC of constants. We call it the Lie algebra of G. In a work in preparation [8], in which he defines "A-group" intrinsically, Kolchin shows that the Lie algebra g of a connected A-group can be given a structure of Received by the editors September 15, 1976. AMS (MOS) subject classifications (1970). Primary 12H05, 17B65.

Canonical Transformations and the Theory of Representations of Lie Groups

Abstract: A new approach is introduced for constructing representations of Lie groups based on canonical transformations in group phase space of the Lie group which reduce the regular representation of the Lie group to canonical form. Examples are considered of the group? (semisimple, nilpotent, solvable) having application to problems in physics, and for these groups formulas are given for the generators, finite rotation operators, characters, and Plancherel measure. The Lie group is viewed as a dynamical system described by a Hamiltonian which is a linear form in the momenta with coefficients depending on the coordinates. This permits the use of the methods of quantum mechanics, in particular integrals of motion, coherent states, and path integrals, for constructing representations of Lie groups.

Initiation of the representation theory of lie admissible algebras of operators on bimodular hilbert spaces

Abstract: In this paper we recall the conventional, one-sided, linear theory of modules; we indicate its equivalence with the one-sided representation theory of algebras; and we point out its applicability to associative, Jordan, and Lie algebras. We then outline the broader, two-sided (left and right), linear theory of modules; we indicate its equivalence with the two-sided (left and right), linear theory of modules; we indicate its equivalence with the two-sided representation theory of algebras; and we point out its applicability also to associative, Jordan, and Lie algebras. For the latter algebras, the methods for the implementation of the conventional one-sided representation theory into the broader two-sided theory are also indicated. Furthermore, we show that the alternative algebras generally admit only the two-sided representation theyr for the linear case. These methods are then applied to the initiation of the representation theory of the Lie-asmissible algebras. We first show that these latter algebras, as for the alternative case, generally admit only the two-sided linear representation theory, as a natural generalization of the two-sided theory of the associative and Lie algebras. A number of properties of the two-sided representation theory of the Lie-admissible algebras are pointed tation theory underlying Heisenberg's equations in quantum mechanicsmore » is in actuality of two-sided character, although as a trivial implementation of the conventional, one-sided theory. We then recall the Lie-admissible algebras are pointed out.« less

Correction to "On the springer representations op the weyl groups of classical algebraic groups"

Abstract: (1979). Correction to 'On the springer representations op the weyl groups of classical algebraic groups' Communications in Algebra: Vol. 7, No. 18, pp. 2027-2033.

Topological Frobenius reciprocity for projective limits of Lie groups

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