Showing papers on "Representation theory published in 1978"

Differential Geometry, Lie Groups, and Symmetric Spaces

Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

Kirillov's character formula for reductive Lie groups

Abstract: Inventiones math. 48, 2007-220 (1978); on-line version. Kirillov’s famous formula says that the characters X of the irreducible unitary representations of a Lie group G should be given by an equation of the form (Φ) χ(exp x )= p(x) −1 Ω e i(λ,x) dµΩ(λ) where ω =Ω (X )i s aG-orbit in the dual g ∗ ft he Lie algebrag of G, µΩ is Kirillov’s canonical measure on Ω, and p is a certain function on g ,n amely p(x )= det 1/2 {sinh(ad(x/2)) /ad(x/2)} at least for generic orbits Ω [10]. This formula cannot be taken too literally, of course (the integral in (Φ) is usually divergent), but has to be interpreted as an equation of distributions on a certain space of test functions on g. To make this precise, denote by g o an open neighborhoodod of zero in g so that exp : g → G restricts to an invertible analytic map of g o onto an open subset of G. For our purposes, the formula (Φ) should be interpreted as saying that (Φ � )t r g ϕ(x)π(exp(x)) dx = Ω g e i(λ,x) ϕ(x) p(x) −1 } dµΩ(λ) for all C ∞ functions ϕ with compact support in g o . (Here π is the representation of G with character χ.) Of course, Kirillov’s formula does not hold in this generality. It is in fact a major problem in representation theory to determine its exact domain of validity. In this paper we shall show that Kirillov’s formula holds for the characters of a reductive real Lie group which occur in the Plancherel formula. Actually, we shall deal in detail only with the discrete series characters. The formula for the other characters can then be reduced to the formula for the discrete series characters by familiar methods. (Duflo [3]). Kirillov’s formula for the discrete series is a consequence of a formula relating the Fourier transform on g with the Fourier transform on Cartan subalgebras of compact type by means of the invariant integral. This is the form in which Kirillov’s formula will be proved.

Polarizations in the Classical Groups

Abstract: So we only consider polarizations of nilpotent elements. A more general definition leads to questions which are usually first reduced to the nilpotent case, cf. the proof of [3] (5.16). The structure of Pol(x) is interesting for the infinite-dimensional representation theory, cf. [7] and [1] (6.3).

Representation theory of artin algebras VI: A functorial approach to aimost split sequences

Abstract: (1978). Representation theory of artin algebras VI: A functorial approach to aimost split sequences. Communications in Algebra: Vol. 6, No. 3, pp. 257-300.

Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces

Abstract: This paper establishes that the homotopy category of rational differential graded commutative coalgebras is equivalent to the homotopy category of rational differential graded Lie algebras which have a nilpotent completion as homology. This generalizes a result which Quillen proved in the simply connected case. When combined with Sullivan's work on rational homotopy theory, our result shows that the homotopy category of rational differential graded Lie algebras with nilpotent finite type homology is equivalent to the rational homotopy category of nilpotent topological spaces with finite type rational homology. Our results include the construction of minimal Lie algebra models for simply connected spaces, and we show that the rational homotopy groups of a simply connected CW complex may be calculated from a free Lie algebra generated by the cells with a differential given on generators by the attaching maps.

The optical group and its subgroups

Abstract: The optical group Opt(3,1) is a ten‐dimensional maximal subgroup of the conformal group of space–time, characterized by the fact that it leaves a lightlike vector subspace in Minkowski space invariant. Thus it is the group underlying the symmetry structure of the parton model in particle physics. The present article is devoted to a complete classification of all closed connected subgroups of Opt(3,1). A list of representatives of all Lie subalgebras of the algebra opt(3,1) is given in the form of tables and many of their properties are established (their invariants, normalizers, isomorphism classes, etc.). Most of the subalgebras of opt(3,1) are also contained in the similitude algebra sim(3,1). We discuss a method for extracting the ’’new’’ subalgebras of opt(3,1) from the list; these will go over into a future list of subalgebras of the conformal Lie algrebra itself.

Unitary representations of Lie groups with cocompact radical and applications

Abstract: The paper gives a necessary and sufficient condition in order that a connected and simply connnected Lie group with cocompact radical be of type I This result is then applied to a characterization of Lie groups, all irreducible unitary representations of which are completely continuous

Simply connected groups, the hyperalgebra, and verma's conjecture.

Abstract: A finite-dimensional representation of a connected affine algebraic group is determined by the behavior of the operators coming from the hyperalgebra of the group. In order to get a complete picture of the representation theory of the group in terms of that of the hyperalgebra, one questions whether every finite-dimensional representation of the hyperalgebra comes from a representation of the group (see [1] for instance). In this paper, we show the following two things: 1. A simply connected group in positive characteristic is the inductive limit of its infinitesimal neighborhoods of the identity. 2. Each finite-dimensional representation of the hyperalgebra of a simply connected group comes from a representation of the group. Each of these two properties characterizes the simply connected groups. The proof of the second fact completes an aspect of the work begun in the paper [1] on the hyperalgebra. In characteristic zero, a group is called simply connected if it cannot be covered by another group by a map with finite, non-trivial kernel. In line with this definition, Takeuchi in [3] defines a group in characteristic p to be simply connected if it has no non-trivial etale coverings. An interpretation of Theorem 1.9 of [3] leads to condition 1 above for simple connectedness in terms of infinitesimal neighborhoods. This condition in turn yields Verma's conjecture: finite-dimensional representations of the hyperalgebra of a simply connected group are rational. Let G be an affine k-group with coordinate ring A over a field k of characteristic p. Let M be the kernel of the augmentation map of A, and let

Representation theory of algebras stably equivalent to an hereditary Artin algebra

Abstract: Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Muller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras. Introduction. We recall that An artin algebra is an artin ring that is a finitely generated module over its center, which is also an artin ring. Let mod A denote the category of finitely generated (left) A-modules, and mod A the category of finitely generated A-modules modulo projectives (see [8]). We also recall that two artin algebras A and A' are said to be stably equivalent if the categories of modules modulo projectives, mod A and mod A', are equivalent. The purpose of this paper is to study the algebras that are stably equivalent to an hereditary artin algebra. This class of algebras contains the artin algebras such that the square of the radical is zero, the hereditary algebras and other algebras that are not hereditary or of radical square zero. We generalize here the results that we proved in [7] for hereditary artin algebras, using the notions of almost split sequences and irreducible maps developed by M. Auslander and I. Reiten. Hereditary artin algebras have also been studied by P. Gabriel, I. Gelfand, Nazarova and Ponomarev, V. Dlab and C. M. Ringel using techniques of representations of diagrams and K-species (see [10], [12]-[14]). These techniques apply also to artin algebras of radical square zero, also studied using different methods by W. Muller (see [15]). The treatment that we do here is quite different from the treatment of the named authors, since it does not rely on diagramatic techniques, but is module theoretical and gives a unified approach to the hereditary and radical square zero cases, as well as to other algebras not considered previously. The ideas Presented to the Society, January 25, 1976; received by the editors September 15, 1976. AMS (MOS) subject classifications (1970). Primary 16A46, 16A64; Secondary 16A62. ? American Mathematical society 1978

The graded Lie groups SU(2,2/1) and OSp(1/4)

Abstract: We study the graded Lie groups corresponding to the graded Lie algebras SU(2,2/1) and OSp(1/4). General finite group transformations are parametrized, and nonlinear representations are obtained on coset spaces. Jordan and traceless algebras are constructed which admit these groups as automorphism groups.

Brauer groups of linear algebraic groups with characters

Abstract: Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)

On Lie algebras of differential formal groups of Ritt

Abstract: 1. Let K be an algebraically closed field of characteristic 0 and let k be a subfield of K. A finite-dimensional vector AT-space D is called (after W. Y. Sit) a Lie espace if D has a structure of a île fc-algebra. Suppose now that D C Der̂ A: and that k = {a G K\\Da = 0}. We denote by K[D] the (associative) algebra of differential operators; it is generated by K and D with relations dk = \\d + d(X), dtd2 d2dx = [dv d2] for d, dv d2GD,\\GK. If M, N are K[D] modules then M ®K N and HomK(M, N) are given natural structures of K[D] modules by dim ® ri) = dm® n + m ® dn and (dy)(m) = d(jp(m)) + B ®K B which is a map of K[D]-algebras. 1.2 DEFINITION. A K[D] -module M is split if M = K ®k M° for some ^-module M° and DM = 0. The split action of d G D will be denoted d°. 1.3 REMARK. If M is a split K[D] -module and we have another action of the elements d G D on M which makes M into a K[D] -module, then d d° G End^Af. For co(d) = d d° we have the relation

Unitary invariance in algebraic algebras

Abstract: ABsTRAcr. A structure theorem is obtained for subspaces invariant under conjugation by the unitary group of a prime algebraic algebra over an infinite field. For an invariant subalgebra W, it is shown that either W is central, W contains an ideal, or the ring satisfies the standard identity of degree eight. Also, for prime algebras not satisfying such an identity, the unitary group is not solvable.